GREATEST COMMON DIVISORS NEAR 𝑆-UNITS, APPLICATIONS, AND
     CONJECTURES ON ARITHMETIC ABELIAN SURFACES
                                 By
                             Zheng Xiao
                        A DISSERTATION
                            Submitted to
                    Michigan State University
            in partial fulfillment of the requirements
                         for the degree of
              Mathematics – Doctor of Philosophy
                                2022


                                           ABSTRACT
We bound the greatest common divisor of two coprime multivariable polynomials evaluated at
algebraic numbers, generalizing the work of Levin by thickening the group of 𝑆-units to allow
for points that are merely “close" to 𝑆-units. Our inequalities make progress towards conjectured
GCD inequalities of Silverman and towards Vojta’s conjecture for blowups. The proofs rely on
Schmidt’s Subspace Theorem.
    As an application, we prove results on the greatest common divisors of terms from two
general linear recurrence sequences, extending the results of Levin, who considered the case
where the linear recurrences are simple. In particular, we improve on recent results of Grieve
and Wang for general linear recurrences, and bound the exceptional set to a logarithmic region.
An example shows that the logarithmic region is necessary.
    On abelian surfaces which come from the Jacobians of hyperelliptic curves, we establish a
connection between GCD conjectures on the abelian surface and conjectures on the arithmetic
discriminant for quadratic points on the associated hyperelliptic curve. It predicts, in particular
situations, a stronger inequality than Vojta’s theorem on the arithmetic discriminant. We give
some examples of extreme values of the arithmetic discriminant.


                                   ACKNOWLEDGEMENTS
As the author of the thesis, I would like to thank Aaron Levin for many helpful comments on
several proofs of the main theorems as well as helping polish the thesis. I would also thank
Nathan Grieve and Joseph Silverman for advice on writing and making the reference list.
    As a PhD student at Michigan State University, I thank all the staff, faculty and colleagues
for creating such a good atmosphere of studying and researching. In particular, I would like to
thank my academic committee members: Aaron Levin, George Pappas, Micheal Shapiro and
Rajesh Kulkarni not only for offering great graduate courses but also for caring me on both
academic and bureaucratic affairs throughout my PhD period.
    In the end I would like to thank my parents for their unconditional trust and support. I would
like to thank my girlfriend, Siqi Wang, for her timely encouragement, endless patience and love
despite the long distance between us. I would like to thank my best friend, Zhihao Zhao, for
his company during all these years. I would like to thank my supervisor, Aaron Levin, for not
just teaching me math, but also teaching me the attitude towards research and starting me up in
my early career. I also thank all my friends, Chuangtian (Armstrong) Guan, Yizhen Zhao, Chen
Zhang, Keping Huang, and all other friends,for giving me a happy and unforgettable experience
at Michigan State University.
                                                iii


                                TABLE OF CONTENTS
CHAPTER 1 INTRODUCTION . .            . . . .  . . . . . . . . . . . . . . . . . . . . . . .  1
   1.1 Diophantine approximation .    . . . .  . . . . . . . . . . . . . . . . . . . . . . .  1
   1.2 Linear recurrence sequences    . . . .  . . . . . . . . . . . . . . . . . . . . . . .  5
   1.3 Arithmetic discriminant . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . .  9
CHAPTER 2 ABSOLUTE VALUES AND HEIGHTS                      . . . . . . . . . . . . . . . . . 12
   2.1 Absolute values . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . 12
   2.2 Height functions on projective spaces . . . . . .   . . . . . . . . . . . . . . . . . 12
   2.3 Height functions on projective varieties . . . .    . . . . . . . . . . . . . . . . . 13
   2.4 Local height functions . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . 15
   2.5 Generalized greatest common divisors . . . . .      . . . . . . . . . . . . . . . . . 17
CHAPTER 3 DIOPHANTINE APPROXIMATION                    . . . . . . . . . . . . . . . . . . . 19
   3.1 Roth’s theorem . . . . . . . . . . . . . . .    . . . . . . . . . . . . . . . . . . . 19
   3.2 Wirsing’s theorem . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . 20
   3.3 Schmidt’s subspace theorem . . . . . . . .      . . . . . . . . . . . . . . . . . . . 21
CHAPTER 4 ARITHMETIC DISCRIMINANT . . . . . . . . . . . . . . . . . . . . .                  22
   4.1 Arithmetic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22
   4.2 Arithmetic discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    23
CHAPTER 5    TOOLS IN LINEAR RECURRENCE SEQUENCES . . . . . . . . . . .                      26
CHAPTER 6 ALMOST 𝑆-UNITS AND ALMOST 𝑆-UNIT EQUATIONS . . . . . . .                           29
   6.1 Compatible definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    29
   6.2 Almost 𝑆-unit equation theorem . . . . . . . . . . . . . . . . . . . . . . . . .      30
CHAPTER 7    PROOFS OF DIOPHANTINE APPROXIMATION THEOREMS . . . .                            32
CHAPTER 8    PROOFS OF LINEAR RECURRENCE SEQUENCES THEOREMS . .                              48
CHAPTER 9    QUADRATIC POINTS ON ABELIAN SURFACES . . . . . . . . . . .                      63
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     66
                                              iv


                                               CHAPTER 1
                                            INTRODUCTION
1.1      Diophantine approximation
Upper bounds for the greatest common divisor of integers of the form 𝑎 𝑛 − 1 and 𝑏 𝑛 − 1 were first
studied by Bugeaud, Corvaja, and Zannier in [3], where they proved the following inequality:
Theorem 1.1.1 (Bugeaud, Corvaja, Zannier [3]). Let 𝑎, 𝑏 be multiplicatively independent
integers, and let 𝜖 > 0. Then, provided 𝑛 sufficiently large, we have
                                     log gcd(𝑎 𝑛 − 1, 𝑏 𝑛 − 1) < 𝜖𝑛.
     Note that even though the statement is simple, the proof requires Schmidt’s Subspace The-
orem from Diophantine approximation. Actually, for most of the following works, it is the
fundamental ingredient in their proofs.
     Corvaja, Zannier [5] and Hernández, Luca [12] subsequently extended Theorem 1.1.1 to
𝑆-unit integers:
Theorem 1.1.2 (Corvaja, Zannier [5] and Hernández, Luca [12]). Let 𝑝 1 , . . . , 𝑝 𝑡 ∈ Z be prime
numbers and let 𝑆 = {∞, 𝑝 1 , . . . , 𝑝 𝑡 }. Then for every 𝜖 > 0,
                           log gcd(𝑢 − 1, 𝑣 − 1) ≤ 𝜖 max{log |𝑢|, log |𝑣|}
for all but finitely many multiplicatively independent 𝑆-unit integers 𝑢, 𝑣 ∈ Z∗𝑆 .
     More generally, Corvaja and Zannier proved an inequality in the case of bivariate polynomi-
als.
Theorem 1.1.3 (Corvaja, Zannier [6]). Let Γ ⊂ G2𝑚 ( Q̄) be a finitely generated group. Let
 𝑓 (𝑥, 𝑦), 𝑔(𝑥, 𝑦) ∈ Q̄[𝑥, 𝑦] be nonconstant coprime polynomials such that not both of them
                                                     1


vanish at (0, 0). For all 𝜖 > 0, there exists a finite union 𝑍 of translates of proper algebraic
subgroups of G2𝑚 such that
                                 log gcd( 𝑓 (𝑢, 𝑣), 𝑔(𝑢, 𝑣)) < 𝜖 max{ℎ(𝑢), ℎ(𝑣)}
for all (𝑢, 𝑣) ∈ Γ \ 𝑍.
    In recent work of Levin [14], the following result was proven, giving an inequality for greatest
common divisors of polynomials evaluated at 𝑆-unit points, which is a higher-dimensional
version of Corvaja-Zannier’s theorem:
Theorem 1.1.4 (Levin [14]). Let 𝑛 be a positive integer. Let Γ ⊂ G𝑛𝑚 ( Q̄) be a finitely generated
group. Let 𝑓 (𝑥1 , . . . , 𝑥 𝑛 ), 𝑔(𝑥1 , . . . , 𝑥 𝑛 ) ∈ Q̄[𝑥 1 , . . . , 𝑥 𝑛 ] be non-constant coprime polynomials
such that not both of them vanish at (0, . . . , 0). Let ℎ(𝛼) denote the (absolute logarithmic)
height of an algebraic number 𝛼. For all 𝜖 > 0, there exists a finite union 𝑍 of translates of
proper algebraic subgroups of G𝑛𝑚 such that
                     log gcd( 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 ), 𝑔(𝑢 1 , . . . , 𝑢 𝑛 )) < 𝜖 max{ℎ(𝑢 1 ), . . . , ℎ(𝑢 𝑛 )}
for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ Γ \ 𝑍.
    In particular, Γ in Theorem 1.1.4 can be taken as the full set of 𝑛-tuples of 𝑆-units in a num-
ber field 𝑘, where 𝑆 is a finite set places of 𝑘 containing the archimedean places. In the above
statement, log gcd is the generalized logarithmic greatest common divisor, which is defined in
Section 2.5.
    In a slightly different direction, Luca [15] extended Theorem 1.1.2 to rational numbers 𝑢 and
𝑣 that are “close" to being an 𝑆-unit. Let 𝑢 be a non-zero rational number, and 𝑆 a fixed finite set
of primes. We may write 𝑢 uniquely, up to a sign, in the form 𝑢 = 𝑢 𝑆 · 𝑢 𝑆¯ , where 𝑢 𝑆 is a rational
number in reduced form having both its numerator and denominator composed of primes in 𝑆,
                                                                    2


and 𝑢 𝑆¯ is a rational number in reduced form having both its numerator and denominator free of
primes from 𝑆. Luca proved the following:
Theorem 1.1.5 (Luca [15]). Let 𝑆 be a finite set of places of Q. For 𝜖 > 0, there exist three
positive constants 𝐾1 , 𝐾2 , 𝐾3 depending on 𝑆 and 𝜖, such that for any rational numbers 𝑢 and 𝑣
satisfying
                               log gcd(𝑢 − 1, 𝑣 − 1) ≥ 𝜖 max{ℎ(𝑢), ℎ(𝑣)},
one of the following three conditions holds:
   (i) max{ℎ𝑟𝑎𝑡 (𝑢), ℎ𝑟𝑎𝑡 (𝑣)} < 𝐾1 ,
  (ii) 𝑢𝑖 = 𝑣 𝑗 with max{|𝑖|, | 𝑗 |} < 𝐾2 ,
                                        ℎ
 (iii) max{ℎ 𝑆¯ (𝑢), ℎ 𝑆¯ (𝑣)} > 𝐾3         ,
                                      log ℎ
                                                          
                                 𝑥              ℎ(𝑥) ℎ(𝑦)
where ℎ 𝑆¯ (𝑢) = ℎ(𝑢 𝑆¯ ), ℎ𝑟𝑎𝑡        = max           ,        and ℎ = max{ℎ(𝑢), ℎ(𝑣)}.
                                 𝑦              ℎ(𝑦) ℎ(𝑥)
     This shows the GCD of two rational integers 𝑢 − 1 and 𝑣 − 1 cannot be large unless 𝑢 and
𝑣 are multiplicatively dependent or have large non-𝑆 height. One main theorem of this thesis
(Corollary 7.0.4) can also be viewed as a generalization of Theorem 1.1.4 along the lines of
Luca’s theorem. It is studied as follows.
     We want to generalize Theorem 1.1.4 beyond the setting of 𝑆-units points. To achieve this
goal, we introduce the definition of almost 𝑆-units: Roughly speaking, an almost (𝑆, 𝛿)-unit for
some set of places 𝑆 in a number field 𝑘 is an element 𝑢 ∈ 𝑘 whose dominant part of its height
is due to an 𝑆-unit.
Definition 1.1.6. For a fixed 𝛿 > 0 and a fixed set of places 𝑆, if 𝑢 ∈ 𝑘 ∗ , then we say 𝑢 is an
almost (𝑆, 𝛿)-unit if
                                                                 
                                              ∑︁                 1
                                  ℎ 𝑆¯ (𝑢) :=     𝜆 𝑣 (𝑢) + 𝜆 𝑣     ≤ 𝛿ℎ(𝑢)
                                              𝑣∉𝑆
                                                                 𝑢
                                                         3


(see Section 2.1 for the definition of 𝜆 𝑣 ). We denote the set of all almost (𝑆, 𝛿)-units by 𝑘 𝑆,𝛿 .
More generally, let
                                       G𝑛𝑚 (𝑘) 𝑆,𝛿 := {u ∈ G𝑛𝑚 (𝑘)|ℎ 𝑆¯ (u) ≤ 𝛿ℎ(u)},
where
                                                                                     
                                                            ∑︁                       1
                                                ℎ 𝑆¯ (u) =         𝜆 𝑣 (u) + 𝜆 𝑣        .
                                                             𝑣∉𝑆
                                                                                     u
    With Definition 1.1.6, we prove the following generalization of Theorem 1.1.4, which shows
that Γ = (O𝑘,𝑆     ∗ ) 𝑛 may be “thickened" to G𝑛 (𝑘)
                                                            𝑚       𝑆,𝛿 for some positive 𝛿 (depending on 𝜖).
Theorem 1.1.7. (Corollary 7.0.7) Let 𝑛 be a positive integer and 𝑘 a number field, 𝑓 (𝑥 1 , . . . , 𝑥 𝑛 ),
𝑔(𝑥 1 , . . . , 𝑥 𝑛 ) ∈ 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] be nonconstant coprime polynomials such that not both of them
vanish at (0, . . . , 0). For all 𝜖 > 0, there exists 𝛿 > 0 and a proper Zariski closed subset 𝑍 of
G𝑛𝑚 such that:
                      log gcd( 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 ), 𝑔(𝑢 1 , . . . , 𝑢 𝑛 )) < 𝜖 max{ℎ(𝑢 1 ), . . . , ℎ(𝑢 𝑛 )}
for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍.
    By Theorem 5 of [8], we may further choose 𝑍 so that it is a (possibly infinite) union of
positive-dimensional torus cosets.
    In fact, we prove the following refinement of Theorem 1.1.7.
Theorem 1.1.8. (Theorem 7.0.6) Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘
containing the archimedean places. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] be coprime polynomials that don’t
both vanish at the origin (0, . . . , 0). For all 0 < 𝛿 < 1, there exists a proper Zariski closed
subset 𝑍 of G𝑛𝑚 such that
                                                                                               𝑛
                                                                                              ∑︁
                                                                                          1/2
                             log gcd( 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 ), 𝑔(𝑢 1 , . . . , 𝑢 𝑛 )) < 𝐶𝛿         ℎ(𝑢𝑖 )
                                                                                              𝑖=1
for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍 satisfying ℎ 𝑆¯ (u) < 𝛿ℎ(u), where 𝐶 = 6(deg 𝑓 + deg 𝑔)𝑛2
is a constant.
                                                                     4


    Theorem 7.0.6 extends Levin’s Theorem 1.1.4 from integral points to rational points, and
may be viewed as progress towards Vojta’s conjecture for certain blown-up varieties. This The-
orem gives a GCD inequality of the form similar to what Vojta’s Conjecture predicts. Assuming
Vojta’s Conjecture, Silverman obtained an upper bound for the polynomial GCD in [18]. By
properly extending the notions from Q to a number field, we can compare Silverman’s conjec-
tural upper bound with our inequality. More precisely, in Remark 7.0.8 we discuss the relation
of Theorem 7.0.6 with conjectured inequalities of Silverman based on Vojta’s conjecture. We
also note work of Grieve [10] in this direction.
1.2    Linear recurrence sequences
On the other hand, Levin [14] also gave a classification (Theorem 1.2.2) of large GCDs among
terms from simple linear recurrence sequences (see also earlier work of Fuchs [9]). A primary
goal of our work is to study the case of general linear recurrences (i.e., without the assumption
that the linear recurrence is simple). In the case of binary linear recurrences, Luca [15] showed:
Theorem 1.2.1 (Luca [15]). Let 𝑎 and 𝑏 be non-zero integers which are multiplicatively inde-
pendent, and let 𝑓 , 𝑔, 𝑓1 and 𝑔1 be non-zero polynomials with integer coefficients. For every
positive integer 𝑛 set
                                           𝑢 𝑛 = 𝑓 (𝑛)𝑎 𝑛 + 𝑔(𝑛)
and
                                          𝑣 𝑛 = 𝑓1 (𝑛)𝑏 𝑛 + 𝑔1 (𝑛).
Then, for every fixed 𝜖 > 0 there exists a positive constant 𝐶𝜖 > 0 depending on 𝜖 and on the
given data 𝑎, 𝑏, 𝑓 , 𝑓1 , 𝑔 and 𝑔1 , such that
                                     log gcd(𝑢 𝑛 , 𝑣 𝑚 ) < 𝜖 max{𝑚, 𝑛}
holds for all pairs of positive integers (𝑚, 𝑛) with max{𝑚, 𝑛} > 𝐶𝜖 .
                                                       5


     The essential assumption is that 𝑎 and 𝑏 are multiplicatively independent integers, which
gives a contradiction to condition (ii) of Theorem 1.1.5. Note that Theorem 1.2.1 is proved
without the assistance of Schmidt’s Subspace Theorem, so one should expect that a stronger
result can be proved with the Subspace Theorem applied to general linear recurrence sequences.
In fact, a recent result due to Grieve and Wang [11] on general linear recurrences generalized
Luca’s binary case, and recovered Levin’s result 1.2.2 at the same time. We will give an alter-
native proof of this theorem later.
     Levin [14] applied Theorem 1.1.4 to terms from simple linear recurrence sequences, giving
a classification of when two such terms may have a large GCD.
Theorem 1.2.2 (Levin [14]). Let
                                                                ∑︁𝑠
                                                   𝐹 (𝑚) =           𝑐𝑖 𝛼𝑖𝑚 ,
                                                                 𝑖=1
                                                                ∑︁𝑡
                                                   𝐺 (𝑛) =           𝑑 𝑗 𝛽𝑛𝑗 ,
                                                                 𝑗=1
define two algebraic simple linear recurrence sequences. Let 𝑘 be a number field such that
𝑐𝑖 , 𝛼𝑖 , 𝑑 𝑗 , 𝛽 𝑗 ∈ 𝑘 for 𝑖 = 1, . . . , 𝑠, 𝑗 = 1, . . . , 𝑡. Let 𝑀𝑘 be the canonical set of places in 𝑘. Let
                         𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Let 𝜖 > 0. All but finitely many solutions (𝑚, 𝑛) of the inequality
                              ∑︁
                                      − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } > 𝜖 max{𝑚, 𝑛}
                            𝑣∈𝑀𝑘 \𝑆0
satisfy one of finitely many linear relations
                                (𝑚, 𝑛) = (𝑎𝑖 𝑡 + 𝑏𝑖 , 𝑐𝑖 𝑡 + 𝑑𝑖 ), 𝑡 ∈ Z, 𝑖 = 1, . . . , 𝑟,
where 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 ∈ Z, 𝑎𝑖 𝑐𝑖 ≠ 0, and the linear recurrences 𝐹 (𝑎𝑖 • +𝑏𝑖 ) and 𝐺 (𝑐𝑖 • +𝑑𝑖 ) have a
nontrivial common factor for 𝑖 = 1, . . . , 𝑟.
                                                                6


    Grieve and Wang [11] have extended Theorem 1.2.2 to general linear recurrence sequences.
Theorem 1.2.3 (Grieve, Wang [11]). Let
                                                ∑︁𝑠
                                       𝐹 (𝑚) =        𝑝𝑖 (𝑚)𝛼𝑖𝑚 ,
                                                𝑖=1
                                                ∑︁ 𝑡
                                        𝐺 (𝑛) =        𝑞 𝑗 (𝑛) 𝛽𝑛𝑗 ,
                                                 𝑗=1
for 𝑛 ∈ N, be algebraic linear recurrence sequences, defined over a number field 𝑘, such that
their roots generate together a torsion-free multiplicative subgroup Γ of 𝑘 × . Suppose that
                                       max{|𝛼𝑖 | 𝑣 , |𝛽 𝑗 | 𝑣 } ≥ 1,
                                        𝑖, 𝑗
for any 𝑣 ∈ 𝑀𝑘 . Let 𝜖 > 0 and consider the inequality
                                log gcd(𝐹 (𝑛), 𝐺 (𝑛)) < 𝜖 max{𝑚, 𝑛}                              (†)
for pairs of positive integers (𝑚, 𝑛) ∈ N2 . The following two assertions hold true.
   1. Consider the case that 𝑚 = 𝑛. If the inequality (†) is valid for infinitely many positive
       integers (𝑛, 𝑛) ∈ N2 , then 𝐹 and 𝐺 have a non-trivial common factor.
   2. Consider the case that 𝑚 ≠ 𝑛. If the inequality (†) is valid for infinitely many pairs of
       positive integers (𝑚, 𝑛) ∈ N2 , with 𝑚 ≠ 𝑛, then the roots of 𝐹 and 𝐺 are multiplicatively
       dependent (see Def 8.0.7). Further, in this case, there exist finitely many pairs of integers
       (𝑎, 𝑏) ∈ Z2 such that
                                       |𝑚𝑎 + 𝑛𝑏| = 𝑜(max{𝑚, 𝑛}).
    The proof of Theorem 1.2.3 in [11] is based on a “moving targets" version of Theorem 1.1.4.
We will give an alternative proof of Theorem 1.2.3 and also give a quantitative improvement
in which the error term 𝑜(max{𝑚, 𝑛}) can be controlled as a constant multiple of log max{𝑚, 𝑛}.
                                                    7


    As an application of the polynomial GCD inequality, we state our main result on linear
recurrence sequences:
Theorem 1.2.4. Let
                                                       ∑︁ 𝑠
                                           𝐹 (𝑚) =            𝑝𝑖 (𝑚)𝛼𝑖𝑚 ,
                                                        𝑖=1
                                                       ∑︁ 𝑡
                                            𝐺 (𝑛) =           𝑞 𝑗 (𝑛) 𝛽𝑛𝑗 ,
                                                         𝑗=1
define two algebraic linear recurrence sequences. Let 𝑘 be a number field such that all coeffi-
cients of 𝑝𝑖 and 𝑞 𝑗 and 𝛼𝑖 , 𝛽 𝑗 are in 𝑘, for 𝑖 = 1, . . . , 𝑠, 𝑗 = 1, . . . , 𝑡. Let
                       𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Then all but finitely many solutions (𝑚, 𝑛) of the inequality:
                            ∑︁
                                   − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } < 𝜖 max{𝑚, 𝑛}
                          𝑣∈𝑀𝑘 \𝑆0
are of the form:
                  (𝑚, 𝑛) = (𝑎𝑖 𝑡, 𝑏𝑖 𝑡) + (𝜇1 , 𝜇2 ),     𝜇1 , 𝜇2 ≪ log 𝑡, 𝑡 ∈ N, 𝑖 = 1, . . . , 𝑟
with finitely many choices of nonzero integers (𝑎𝑖 , 𝑏𝑖 ) .
Moreover, if the roots of 𝐹 and 𝐺 are independent (see Def 8.0.7), then the solutions (𝑚, 𝑛)
satisfy one of the finitely many linear relations:
                              (𝑚, 𝑛) = (𝑎𝑖 𝑡 + 𝑏𝑖 , 𝑐𝑖 𝑡 + 𝑑𝑖 ), 𝑡 ∈ N, 𝑖 = 1, . . . , 𝑟
where 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 ∈ N, 𝑎𝑖 𝑐𝑖 ≠ 0, and the linear recurrences 𝐹 (𝑎𝑖 • +𝑏𝑖 ) and 𝐺 (𝑐𝑖 • +𝑑𝑖 ) have a
nontrivial common factor for 𝑖 = 1, . . . , 𝑟.
Example 1.2.5. Under the set up of Theorem 1.2.4, we give an example illustrating the necessity
of (𝜇1 , 𝜇2 ) in the statement:
                                                           8


     Define the two linear recurrence sequences as: 𝐹 (𝑚) = 𝑚 𝑝 𝑚 + 1, 𝐺 (𝑛) = 𝑝 𝑛 + 1, where 𝑝
is a prime. In the notations of Theorem 1.2.4, 𝑆0 = ∅. Let 𝜖 < log 2. It is easily seen that for
                                                  𝑘 +𝑘
(𝑚, 𝑛) = ( 𝑝 𝑘 , 𝑝 𝑘 + 𝑘), ∀𝑘 ∈ Z>0 , 𝐹 (𝑚) = 𝑝 𝑝       + 1 = 𝐺 (𝑛), so the inequality
                                                     𝑘 +𝑘
              log gcd{|𝐹 (𝑚)|, |𝐺 (𝑛)|} = log( 𝑝 𝑝        + 1) > 𝜖 ( 𝑝 𝑘 + 𝑘) = 𝜖 max{𝑚, 𝑛}
holds for infinitely many 𝑘 and hence infinitely many (𝑚, 𝑛). It is easily seen that such pairs
(𝑚, 𝑛) do not lie on finitely many lines, but do lie in a logarithmic region around the line 𝑥 = 𝑦,
i.e., for such pairs we may write (𝑚, 𝑛) = (𝑡, 𝑡) + (𝜇1 , 𝜇2 ) with 𝜇1 , 𝜇2 ≪ log 𝑡 in agreement with
Theorem 1.2.4.
1.3      Arithmetic discriminant
Vojta defined the arithmetic discriminant 𝑑 𝑎 in the proofs of [23] and [21], and an alternative
definition is given in the other paper [22] under arithmetic geometry. In [23], he obtained a first
estimate of 𝑑 𝑎 . Later, Vojta successfully proved the Vojta’s conjecture [24] replacing 𝑑 (𝑃), the
usual logarithmic discriminant, by 𝑑 𝑎 (𝑃) in the curve case.
Theorem 1.3.1 (Theorem 4.2.4). Let 𝐶 be a curve over a numerb field 𝑘 and let 𝜋 : 𝑋 → 𝐵
be a regular model, where 𝐵 is the arithmetic curve corresponding to SpecO 𝑘 . Fix an integer
𝜈 ≥ 1, a real number 𝜖 > 0, an effective divisor 𝐷 on 𝑋 with no multiple components, and a
divisor 𝐴 on 𝑋 which is ample on the generic fibre. Then for all points 𝑃 ∈ 𝐶 (𝑘) \ Supp(𝐷)
with [𝑘 (𝑃) : 𝑘] ≤ 𝜈,
                            𝑚(𝐷, 𝑃) + ℎ 𝐾 (𝑃) ≤ 𝑑 𝑎 (𝑃) + 𝜖 ℎ 𝐴 (𝑃) + 𝑂 (1),
where the constant in 𝑂 (1) depends on 𝑋, 𝐷, 𝜈, 𝐴 and 𝜖.
     The proof gives a deep and extraordinary construction, which somehow has the similar
flavor of the proof of Roth’s theorem. However this result is not applied widely and needs more
attention.
                                                     9


     A follow up theorem, proven in [22], gives a finiteness statement for points of bounded
degree on curves.
Theorem 1.3.2. Let 𝑓 : 𝐶 → P1 be a dominant morphism, let 𝑠 ∈ N, and let 𝑔 be the genus of
𝐶. Assume also that
                                       𝑔 − 1 > (deg 𝑓 )(𝑠 − 1).
Then the set
                                   ¯
                          {𝑃 ∈ 𝐶 ( 𝑘)|[𝑘 (𝑃) : 𝑘] ≤ 𝑠 and 𝑘 ( 𝑓 (𝑃)) = 𝑘 (𝑃)}
is finite.
     Using well-known upper bounds for the gonality of a curve, one immediately finds,
Corollary 1.3.3. If 𝑔 ≥ 6 then there exists a dominant morphism 𝑓 : 𝐶 → P1 such that the set
                                   ¯
                          {𝑃 ∈ 𝐶 ( 𝑘)|[𝑘 (𝑃) : 𝑘] ≤ 2 and 𝑘 ( 𝑓 (𝑃)) = 𝑘 (𝑃)}
is finite.
     Song-Tucker [19] gave a general version of Theorem 1.3.2.
Proposition 1.3.4. Let 𝐶 and 𝐶 ′ be curves of genus 𝑔 and 𝑔′, respectively, defined over a number
field 𝑘, let 𝜈 be a positive integer, and let 𝑓 : 𝐶 → 𝐶 ′ be a dominant 𝑘-morphism. Assume that
                                      𝑔 − 1 > (𝜈 + 𝑔′ − 1) deg 𝑓 .
Then the set
                                   ¯
                          {𝑃 ∈ 𝐶 ( 𝑘)|[𝑘 (𝑃) : 𝐾] = 𝜈 and 𝑘 ( 𝑓 (𝑃)) = 𝑘 (𝑃)}
is finite.
     As an application, they obtained a stronger Castelnuovo’s genus inequality under certain
conditions.
                                                   10


    In here, we continue study the connection between the arithmetic discriminant and the GCD
conjecture, giving an equivalence of conjectures for quadratic points.
    In later chapters, we will give the proofs of the main Diophantine approximation results and
the application to linear recurrence sequences, respectively. We will also develope the connection
between the GCD conjecture and the conjecture on arithmetic discriminant on quadratic points.
                                                 11


                                                  CHAPTER 2
                              ABSOLUTE VALUES AND HEIGHTS
2.1     Absolute values
Let 𝑘 be a number field, 𝑀𝑘 the set of places of 𝑘 and O𝑘 the ring of integers of 𝑘. For 𝑣 ∈ 𝑀𝑘 ,
let 𝑘 𝑣 denote the completion of 𝑘 with respect to 𝑣. Throughout the thesis, we normalize the
absolute value | · | 𝑣 corresponding to 𝑣 ∈ 𝑀𝑘 as follows: If 𝑣 is archimedean and 𝜎 is the
corresponding embedding 𝜎 : 𝑘 → C, then for 𝑥 ∈ 𝑘 ∗ , |𝑥| 𝑣 = |𝜎(𝑥)| |𝑘 𝑣 :R|/|𝑘:Q| ; if 𝑣 is non-
archimedean corresponding to a prime ideal 𝒫 in O𝑘 which lies above a rational prime 𝑝, then
it is normalized so that | 𝑝| 𝑣 = 𝑝 −|𝑘 𝑣 :Q 𝑝 |/|𝑘:Q| . In this notation, we have the product formula:
                                                    Ö
                                                          |𝑥| 𝑣 = 1
                                                   𝑣∈𝑀𝑘
for all 𝑥 ∈ 𝑘 ∗ .
     Let 𝑆 be a finite set of places in 𝑀𝑘 . The ring of 𝑆-integers and the group of 𝑆-units are
denoted by O𝑘,𝑆 and O𝑘,𝑆 ∗ respectively.
2.2     Height functions on projective spaces
We will define height functions as in [13]. For a point 𝑃 = (𝛼0 : 𝛼1 : · · · : 𝛼𝑛 ) ∈ P𝑛 (𝑘), we
define its height to be
                                            ∑︁
                                ℎ(𝑃) =             log max{|𝛼0 | 𝑣 , . . . , |𝛼𝑛 | 𝑣 }
                                           𝑣∈𝑀𝑘
and for any 𝑥 ∈ 𝑘, its height ℎ(𝑥) is defined to be the height of the point (1 : 𝑥) in P1 (𝑘).
Lemma 2.2.1. We have the following properties of height functions:
    1. The height ℎ(𝑃) is independent of the choice of homogeneous coordinates for 𝑃.
                                                          12


    2. ℎ(𝑃) ≥ 0 for all 𝑃 ∈ P𝑛 (𝑘).
Proposition 2.2.2. The action of the Galois group on P𝑛 ( Q̄) leaves the height invariant.
     The following finiteness theorem is of fundamental importance for the application of height
functions in Diophantine geometry.
Theorem 2.2.3 (Northcott property). For any numbers 𝐵, 𝐷 ≥ 0, the set
                              {𝑃 ∈ P𝑛 ( Q̄)|ℎ(𝑃) ≤ 𝐵, [Q(𝑃) : 𝑄] ≤ 𝐷}
is finite. In particular, for any fixed number field 𝑘, the set
                                        {𝑃 ∈ P𝑛 (𝑘)|ℎ(𝑃) ≤ 𝐵}
is finite.
2.3      Height functions on projective varieties
Weil’s “Height Machine" constructs a height function associated to every divisor on a projective
variety. These height functions satisfy the following properties.
Theorem 2.3.1. Let 𝑘 be a number field. For every smooth projective variety 𝑉/𝑘 there exists
a map
                                ℎ𝑉 : Div(𝑉) → {functions 𝑉 ( 𝑘¯ → R)}
with the following properties:
    1. (Normalization) Let 𝐻 ⊂ P𝑛 be a hyperplane, and let ℎ(𝑃) be the absolute logarithmic
        height on P𝑛 . Then
                                           ℎP𝑛 ,𝐻 (𝑃) = ℎ(𝑃) + 𝑂 (1)
                         ¯
        for all 𝑃 ∈ P𝑛 ( 𝑘).
                                                    13


2. (Functoriality) Let 𝜙 : 𝑉 → 𝑊 be a morphism and let 𝐷 ∈ Div(𝑊). Then
                               ℎ𝑉,𝜙∗ 𝐷 (𝑃) = ℎ𝑊,𝐷 (𝜙(𝑃)) + 𝑂 (1),
                   ¯
   for all 𝑃 ∈ 𝑉 ( 𝑘)
3. (Additivity) Let 𝐷, 𝐸 ∈ Div(𝑉). Then
                            ℎ𝑉,𝐷+𝐸 (𝑃) = ℎ𝑉,𝐷 (𝑃) + ℎ𝑉,𝐸 (𝑃) + 𝑂 (1)
                   ¯
   for all 𝑃 ∈ 𝑉 ( 𝑘).
4. (Linear Equivalence) Let 𝐷, 𝐸 ∈ Div(𝑉) with 𝐷 linearly equivalent to 𝐸. Then
                                  ℎ𝑉,𝐷 (𝑃) = ℎ𝑉,𝐸 (𝑃) + 𝑂 (1)
                   ¯
   for all 𝑃 ∈ 𝑉 ( 𝑘).
5. (Positivity) Let 𝐷 ∈ Div(𝑉) be an effective divisor, and let 𝐵 be the base locus of the
   linear system |𝐷|. Then
                                          ℎ𝑉,𝐷 (𝑃) ≥ 𝑂 (1)
                        ¯
   for all 𝑃 ∈ (𝑉 \ 𝐵)( 𝑘).
6. (Algebraic Equivalence) Let 𝐷, 𝐸 ∈ Div(𝑉) with 𝐷 ample and 𝐸 algebraically equivalent
   to 0. Then
                                                    ℎ𝑉,𝐸 (𝑃)
                                         lim                 = 0.
                                      ¯ 𝑉 ,𝐷 (𝑃)→∞
                                𝑃∈𝑉 ( 𝑘),ℎ          ℎ𝑉,𝐷 (𝑃)
7. (Finiteness) Let 𝐷 ∈ Div(𝑉) be ample. Then for every finite extension 𝑘 ′/𝑘 and every
   constant 𝐵, the set
                                   {𝑃 ∈ 𝑉 (𝑘 ′)|ℎ𝑉,𝐷 (𝑃) ≤ 𝐵}
   is finite.
8. The height functions ℎ𝑉,𝐷 are determined, up to 𝑂 (1), by normalization, functoriality for
   embeddings 𝜙 : 𝑉 → P𝑛 , and additivity.
                                              14


2.4     Local height functions
We now define local height functions. Let 𝑉 be a projective variety over a number field 𝑘.
Let 𝐷 be a Cartier divisor on 𝑉 and 𝑣 ∈ 𝑀𝑘 . First we define the support of a Cartier divisor
𝐷 = (𝑈𝛼 , 𝑓𝛼 )𝛼∈𝐼 to be
                                              Ø
                                                                  ∗
                                 supp(𝐷) :=       {𝑥 ∈ 𝑈𝛼 | 𝑓𝛼 ∉ O𝑉,𝑥 },
                                               𝛼
where     ∗
        O𝑉,𝑥 is the group of units in the local ring O𝑉,𝑥 . For notation convenience, we write
                                         𝑉𝐷 = 𝑉 \ supp(𝐷)
for the complement of the support of 𝐷. We would like to associate to each place 𝑣 ∈ 𝑀𝑘 a
function
                                         𝜆 𝐷,𝑣 : 𝑉𝐷 (𝑘 𝑣 ) → R
so that the sum
                                             ∑︁
                                                 𝜆 𝐷,𝑣 = ℎ 𝐷
                                            𝑣∈𝑀𝑘
for all points in 𝑉𝐷 (𝑘). Moreover, the local height functions should be additive in 𝐷. If 𝐷 is a
prime divisor, then 𝜆 𝐷,𝑣 should be geometric in the following intuitive sense
                          𝜆 𝐷,𝑣 (𝑃) = − log(𝑣-adic distance from 𝑃 to 𝐷).
To make things precise, we need some definitions. We define an 𝑀𝑘 -constant to be a map
                                             𝛾 : 𝑀𝑘 → R
with the property that 𝛾(𝑣) = 0 for all but finitely many 𝑣 ∈ 𝑀𝑘 . We say that a real-valued
function 𝜙 on a subset 𝑌 of 𝑉 (𝑘) × 𝑀𝑘 is 𝑀𝑘 -bounded if there is an 𝑀𝑘 constant 𝛾 such that
                                           |𝜙(𝑃, 𝑣)| ≤ 𝛾(𝑣)
for all (𝑃, 𝑣) ∈ 𝑌 . We will write 𝑂 𝑣 (1) for an 𝑀𝑘 -bounded function. We say a subset 𝑌 of
𝑉 (𝑘) × 𝑀𝑘 is affine 𝑀𝑘 -bounded if there is an affine open subset 𝑉0 of 𝑉 with affine coordinates
                                                   15


𝑥 1 , . . . , 𝑥 𝑛 such that 𝑌 ⊂ 𝑉0 × 𝑀𝑘 and such that the function
                                   𝑉0 (𝑘) × 𝑀𝑘 → R, 𝑃 ↦→ max |𝑥𝑖 (𝑃)| 𝑣 ,
                                                                     1≤𝑖≤𝑛
is 𝑀𝑘 -bounded on 𝑌 . We say the set 𝑌 is 𝑀𝑘 -bounded if it is a finite union of affine 𝑀𝑘 -bounded
sets. We are ready to state the local height machine. For 𝑣 ∈ 𝑀𝑘 , let 𝑣(𝑥) = − log |𝑥| 𝑣 .
Theorem 2.4.1. Let 𝑉/𝑘 be a smooth projective variety. For each 𝐷 ∈ Div(𝑉) it is possible to
assign a function
                                                     Þ
                                           𝜆𝐷 :           𝑉𝐷 (𝑘 𝑣 ) → R,
                                                    𝑣∈𝑀𝑘
called the local height function with respect to 𝐷, such that the following properties hold:
     1. (Normalization) Let 𝑓 ∈ 𝑘 (𝑉) ∗ be a rational function on 𝑉, and let 𝐷 = div( 𝑓 ) be the
           divisor of 𝑓 . Then the difference
                                                    𝜆 𝐷,𝑣 (𝑃) − 𝑣( 𝑓 (𝑃))
           is an 𝑀𝑘 bounded function on every 𝑀𝑘 bounded subset of 𝑉𝐷 (𝑘) × 𝑀𝑘 .
     2. (Additivity) For all 𝐷 1 , 𝐷 2 ∈ Div(𝑉),
                                         𝜆 𝐷 1 +𝐷 2 ,𝑣 = 𝜆 𝐷 1 ,𝑣 + 𝜆 𝐷 2 ,𝑣 + 𝑂 𝑣 (1).
     3. (Functoriality) Let 𝜙 : 𝑉 → 𝑊 be a morphism of smooth varieties. Then
                                              𝜆 𝜙∗ 𝐷,𝑣 = 𝜆 𝐷,𝑣 ◦ 𝜙 + 𝑂 𝑣 (1).
     4. (Positivity) Let 𝐷 ≥ 0 be an effective divisor. Then
                                                       𝜆 𝐷,𝑣 ≥ 𝑂 𝑣 (1).
     5. (Local/Global Property) Let 𝐷 ∈ Div(𝑉), and let ℎ 𝐷 be a Weil height attached to 𝐷. Then
                                                          ∑︁
                                           ℎ 𝐷 (𝑃) =             𝜆 𝐷,𝑣 (𝑃) + 𝑂 (1)
                                                         𝑣∈𝑀𝑘
           for all 𝑃 ∈ 𝑉𝐷 (𝑘).
                                                           16


    In particular, if 𝐷 is a hypersurface in P𝑛 given by a homogeneous polynomial 𝐹 (𝑥 0 , . . . , 𝑥 𝑛 ) =
0 of degree 𝑑, we have a choice of local height function
                                                            |𝛼𝑖 | 𝑣𝑑            |𝑃| 𝑣𝑑
                              𝜆 𝐷,𝑣 (𝑃) = log max                      = log
                                              𝑖=0,...,𝑛  |𝐹 (𝑃)| 𝑣            |𝐹 (𝑃)| 𝑣
where 𝑃 is written in coordinates (𝛼0 : · · · : 𝛼𝑛 ) ∈ P𝑛 (𝑘) \ Supp(𝐷) and |𝑃| 𝑣 = max𝑖 |𝛼𝑖 | 𝑣 .
For any 𝑥 ∈ 𝑘 and 𝑣 ∈ 𝑀𝑘 , we define the local height of 𝑥 with respect to 𝑣 to be 𝜆 𝑣 (𝑥) =
log max{1, |𝑥| 𝑣 }.
    For a point 𝑃 = (𝛼1 , · · · , 𝛼𝑛 ) ∈ G𝑛𝑚 (𝑘) and a place 𝑣 ∈ 𝑀𝑘 , we define its height and local
                       ∑︁
height as ℎ(𝑃) =           log max{1, |𝛼1 | 𝑣 , . . . , |𝛼𝑛 | 𝑣 } and 𝜆 𝑣 (𝑃) = log max{1, |𝛼1 | 𝑣 , . . . , |𝛼𝑛 | 𝑣 },
                      𝑣∈𝑀𝑘
respectively.
    For a finite set of places 𝑆, we define the proximity function associated to 𝐷 to be the
                                                            ∑︁
                                        𝑚(𝐷, 𝑆, 𝑃) =               𝜆 𝐷,𝑣 (𝑃).
                                                            𝑣∈𝑆
2.5    Generalized greatest common divisors
One can extend the notion of log gcd(𝑎, 𝑏) to all algebraic numbers. Note that for 𝑎 and 𝑏
integers, we calculate their greatest common divisor as:
                                             ∑︁
                         log gcd(𝑎, 𝑏) =              min{ord 𝑝 (𝑎), ord 𝑝 (𝑏)} log 𝑝
                                            𝑝 prime
                                                 ∑︁
                                         =−               log max{|𝑎| 𝑣 , |𝑏| 𝑣 }
                                              𝑣∈𝑀Q,fin
                                                 ∑︁
                                         =−               log− max{|𝑎| 𝑣 , |𝑏| 𝑣 }
                                              𝑣∈𝑀Q,fin
where 𝑀Q,fin is the set of nonarchimedean places of Q and log− 𝑧 = min{0, log 𝑧}. Similarly we
define log+ 𝑧 = max{0, log 𝑧}. With this observation, by adding contributions of archimedean
places, the generalized greatest common divisor is defined as:
                                                         17


Definition 2.5.1. Let 𝑎, 𝑏 ∈ Q̄ be two algebraic numbers, not both zero. We define the
generalized logarithmic greatest common divisors of 𝑎 and 𝑏 by
                                            ∑︁
                         log gcd(𝑎, 𝑏) = −       log− max{|𝑎| 𝑣 , |𝑏| 𝑣 }
                                           𝑣∈𝑀𝑘
where 𝑘 is any number field containing both 𝑎 and 𝑏.
   We will work with this generalized definition in the following chapters.
                                              18


                                          CHAPTER 3
                             DIOPHANTINE APPROXIMATION
3.1    Roth’s theorem
The fundamental problem in Diophantine approximation is how closely an irrational number
can be approximated by a rational number. Precisely, let 𝑎 ∈ R be a given real number, and let
𝑒 > 0 be a given exponent. We ask whether or not the inequality
                                           𝑝           1
                                              −𝑎 ≤ 𝑒
                                           𝑞          𝑞
can have infinitely many solutions in rational numbers 𝑝/𝑞 ∈ Q.
    Dirichlet, in 1842, showed that we can find rational numbers that are fairly close to a given
real number.
Proposition 3.1.1 (Dirichlet). Let 𝑎 ∈ R with 𝑎 ∉ Q. Then there are infinitely many rational
numbers 𝑝/𝑞 ∈ Q satisfying
                                           𝑝          1
                                             − 𝑎 ≤ 2.
                                           𝑞          𝑞
    Next result, due to Liouville in 1844, gives an estimate in the other direction.
Proposition 3.1.2 (Liouville). Let 𝑎 ∈ Q̄ be an algebraic number of degree 𝑑 = [Q(𝑎) : Q] ≥ 2.
Fix a constant 𝜖 > 0. Then there are only finitely many rational numbers 𝑝/𝑞 ∈ Q satisfying
                                            𝑝       1
                                               ≤ 𝑑+𝜖 .
                                            𝑞     𝑞
                                                                              √
    Later, Thue (1909) improved 𝑑 to 𝑑/2 + 1, Siegel (1921) improved to 2 𝑑, Gelfand, Dyson
                     √
(1947) improved to 2𝑑, and finally Roth (1955) improved to 2. In fact, the exponent 2 + 𝜖 is
essentially best possible since we already have Dirichlet’s result.
                                                19


Theorem 3.1.3 (Roth). For every algebraic number 𝑎 and every 𝜖 > 0, the inequality
                                          𝑝             1
                                            − 𝑎 ≤ 2+𝜖
                                          𝑞           𝑞
has only finitely many rational solutions 𝑝/𝑞 ∈ Q.
     A general formulation of Roth’s theorem is the following.
Theorem 3.1.4. Let 𝑘 be a number field, let 𝑆 ⊂ 𝑀𝑘 be a finite set of places on 𝑘, and assume
that each place extends in some way to 𝑘. ¯ Let 𝑎 ∈ 𝑘¯ and 𝜖 > 0 be given. Then there are only
finitely many 𝑏 ∈ 𝑘 satisfying the inequality
                                Ö                              1
                                    min{|𝑏 − 𝑎| 𝑣 , 1} ≤              .
                                𝑣∈𝑆
                                                          𝐻 𝑘 (𝑏) 2+𝜖
     As an application, we have Siegel’s famous theorem on integral points on curves [16]. Let
𝐶 be a geometrically irreducible affine curve over a number field 𝑘 and let 𝑆 be a finite set of
places containing archimedean places. We assume that 𝐶 is given as a closed subvariety of A𝑛𝑘 .
     Let 𝜋 : 𝐶˜aff → 𝐶 be the normalization of 𝐶 and we extend the affine curve 𝐶˜aff to a smooth
projective curve 𝐶, ˜ which is unique up to isomorphism. The points in 𝐶˜ \ 𝐶˜aff are called the
points of 𝐶 at ∞.
     Then Siegel’s theorem on integral points on curves states:
Theorem 3.1.5 (Siegel). If 𝐶˜ has genus 𝑔 > 0 or 𝐶 has at least three distinct points at ∞, then
𝐶 has only finitely many 𝑆-integral points.
3.2     Wirsing’s theorem
Instead of taking the approximating elements from a fixed number field, another direction to
generalize Roth’s theorem is to consider approximation by algebraic numbers of bounded degree.
Toward this end, Wirsing [25] proved a generalization of Roth’s theorem, which we state in a
general form.
                                                20


Theorem 3.2.1 (Wirsing). Let 𝑆 be a finite set of places of a number field 𝑘. Let 𝑃1 , . . . , 𝑃𝑞 ∈
P1 (𝑘) be distinct points and let 𝐷 = 𝑖=1 𝑃𝑖 . Let 𝜖 > 0 and let 𝑑 be a positive integer. Then for
                                                Í𝑞
all but finitely many points 𝑃 ∈ P1 ( 𝑘)      ¯ \ Supp𝐷 satisfying [𝑘 (𝑃) : 𝑘] ≤ 𝑑, we have
                                          𝑚(𝐷, 𝑆, 𝑃) < (2𝑑 + 𝜖)ℎ(𝑃).
Remark 3.2.2. When 𝑑 = 1, Wirsing’s theorem recovers Roth’s theorem. It itself is also a
special case of Vojta’s inequality of arithmetic discriminant (Theorem 4.2.4), with 𝑔 = 0 and 𝜈
arbitrary.
3.3     Schmidt’s subspace theorem
A powerful tool in Diophantine Approximation is the famous Schmidt’s Subspace Theorem,
which will be the primary tool used in the proofs of this thesis.
Theorem 3.3.1 (Schmidt’s Subspace Theorem). Let 𝑘 be a number field and 𝑆 ⊂ 𝑀𝑘 a finite
set of places, 𝑛 ∈ N and 𝜖 > 0. For every 𝑣 ∈ 𝑆, let {𝐿 0𝑣 , . . . , 𝐿 𝑛𝑣 } be a linearly independent set
of linear forms in the variables 𝑥 0 , . . . , 𝑥 𝑛 with coefficients in 𝑘. Then there are finitely many
hyperplanes 𝑇1 , . . . , 𝑇ℎ of P𝑛𝑘 such that the set of solutions x = (𝑥0 : . . . : 𝑥 𝑛 ) ∈ P𝑛𝑘 (𝑘) of
                                         𝑛
                               ∑︁      Ö          |x| 𝑣
                                   log                    ≥ (𝑛 + 1 + 𝜖)ℎ(x) + 𝑂 (1)
                               𝑣∈𝑆      𝑖=0
                                              |𝐿 𝑖 (x)| 𝑣
                                                 𝑣
is contained in 𝑇1 ∪ · · · ∪ 𝑇ℎ . If we take 𝐷 𝑣 to be the sum of divisors defined by 𝐿 𝑖𝑣 , 𝑖 = 0, . . . , 𝑛
and let 𝐾P𝑛 be the canonical divisor of P𝑛 , then this inequality can be written as
                                   ∑︁
                                       𝜆 𝐷 𝑣 ,𝑣 (x) + ℎ 𝐾P𝑛 (x) ≥ 𝜖 ℎ(x) + 𝑂 (1).
                                   𝑣∈𝑆
    If 𝐻𝑖𝑣 is the hyperplane defined by 𝐿 𝑖𝑣 , then the left-hand side of the inequality may be written
    ∑︁ ∑︁ 𝑛
as            𝜆 𝐻𝑖𝑣 ,𝑣 (𝑥) up to 𝑂 (1).
   𝑣∈𝑀𝑘 𝑖=0
Remark 3.3.2. A direct application of the Subspace Theorem is the unit equation. We will give
a more general version of it later in 6.2.2.
                                                           21


                                           CHAPTER 4
                                ARITHMETIC DISCRIMINANT
4.1     Arithmetic varieties
In this section, we will deal with arithmetic objects and we will follow Vojta’s [22] notations
under the general settings of Arakelov geometry.
Definition 4.1.1. An arithmetic variety X consists of the following:
   1. The finite part is a reduced scheme Xfin which is projective and flat over SpecZ and whose
       generic fibre is smooth. Write X∞ = Xfin ×SpecZ SpecC.
   2. The arithmetic part of X consists of a smooth function
                                       ΛX = Λ : X∞ × X∞ → R≥0 ,
       such that Λ(𝑃1 , 𝑃2 ) = 0 if and only if 𝑃1 = 𝑃2 ,
                                         Λ(𝑃1 , 𝑃2 ) = Λ(𝑃2 , 𝑃1 ),
       and
                      Λ(𝑃1 , 𝑃2 ) ≫≪ |𝑧1 (𝑃1 ) − 𝑧1 (𝑃2 )| 2 + · · · + |𝑧 𝑛 (𝑃1 ) − 𝑧 𝑛 (𝑃2 )| 2
       in a neighborhood of the diagonal, where 𝑧1 , . . . , 𝑧 𝑛 are local coordinates on some open
       subset of X∞ . Thus Λ will be called a distance function. We also require that the form
                                            𝜔 := 𝑑1 𝑑1𝑐 Λ(𝑃, 𝑃)
       be a K¥ahler form on X∞ (where 𝑑1 and 𝑑1𝑐 apply only to the first coordinate). Moreover,
       let
                                     𝜆(𝑃, 𝑄) = − log Λ(𝑃, 𝑄), 𝑃 ≠ 𝑄.
                                                 22


    A morphism 𝑓 of arithmetic varieties is a morphism 𝑓fin of their finite parts. Arithmetic
curves and arithmetic surfaces are arithmetic varieties of relative dimension zero and one,
respectively.
    For a number field 𝑘, let 𝑅 be its ring of integers, and let 𝐵 be the arithmetic scheme
with 𝐵fin = Spec𝑅 and Λ𝐵 (𝜎, 𝜏) = 0 if 𝜎 = 𝜏 and 1 otherwise, with 𝜎, 𝜏 ∈ 𝐵∞ . Note that
𝐵∞ = {𝜎 : 𝑘 ↩→ C}. We define the arithmetic curve corresponding to 𝑅 as the arithmetic
curve obtained by using this choice of Λ. An arithmetic variety X over 𝐵 is an arithmetic
variety X, together with a morphism 𝜋 : X → 𝐵. For 𝜎 : 𝑘 ↩→ C, let X𝜎 = Xfin ×𝜎 C, so
            Ý
that X∞ = 𝜎∈𝐵∞ X𝜎 . Also, we may refer to fibres of 𝜋 𝑓 𝑖𝑛 as (non-archimedean) fibres of 𝜋.
More generally, one should view an arithmetic scheme as a scheme with an additional fibre
over the archimedean absolute value of Q. Therefore we inherit the notions of local rings and
(non-archimedean and generic) fibres from Xfin .
    We also have the arithmetic version of divisors and sheaves.
Definition 4.1.2. An arithmetic divisor 𝐷 on X is a divisor 𝐷 fin on Xfin , together with a smooth
function 𝑔 𝐷 : X∞ \ |𝐷 ∞ | → R, such that if for all open sets 𝑈 ⊂ X∞ on which 𝐷 ∞ := 𝐷 fin |𝑈 is
locally represented by a function 𝑓 , the function
                             𝑔 𝐷 (𝑃) + log | 𝑓 (𝑃)| 2 , 𝑃 ∉ Supp(𝐷 ∞ )
extends to a continuous function of 𝑃 on all of 𝑈.
Definition 4.1.3. An invertible sheaf L on X is an invertible sheaf Lfin on Xfin , provided with a
metric on L∞ := Lfin | X∞ compatible with the action of complex conjugation.
4.2    Arithmetic discriminants
            ¯ be an algebraic point on a regular arithmetic surface 𝑋. Let 𝐹 = 𝑘 (𝑃). Let 𝐵 be
Let 𝑃 ∈ 𝑋 ( 𝑘)
the arithmetic curve corresponding to Spec(O𝐹 ), and let 𝑖 : 𝐵 → 𝐸 𝑃 be the prime horizontal
divisor corresponding to 𝑃.
                                                 23


Definition 4.2.1. The arithmetic discriminant 𝑑 𝑎 (𝑃) of 𝑃 on 𝑋 is defined by the formula
                                                    deg 𝑖 ∗ Ω𝐸 𝑃 /𝐵
                                         𝑑 𝑎 (𝑃) =                  .
                                                       [𝐹 : Q]
     An alternative characterization of 𝑑 𝑎 is the following.
Definition 4.2.2. Let 𝐾 𝑋/𝐵 be a divisor corresponding to 𝜔 𝑋/𝐵 . Then we define
                                                  (𝐸 𝑃 .𝐸 𝑃 + 𝐾 𝑋/𝐵 )
                                     𝑑 𝑎 (𝑃) :=                       .
                                                        [𝐹 : Q]
     Note that under this sense, we could write height functions as
                                                        (𝐸 𝑃 .𝐷)
                                            ℎ 𝐷 (𝑃) =             .
                                                        [𝐹 : Q]
     The arithmetic discriminant was initially defined in [23] or [21] using the alternative def-
inition. It is clear that in the function field case, 𝑑 𝑎 is a function of the arithmetic genus
𝑝 𝑎 (𝐸 𝑃 ):
                                           2𝑝 𝑎 (𝐸 𝑃 ) − 2
                               𝑑 𝑎 (𝑃) =                    − (2𝑔(𝐵) − 2).
                                               [𝐹 : 𝑘]
Compared to the discriminant defined in [20],
                                         log |𝐷 𝐹/Q | deg ΩNor(𝐸 𝑃 )/𝐵
                               𝑑 (𝑃) =                  =                ,
                                            [𝐹 : Q]            [𝐹 : Q]
where Nor(𝐸 𝑃 ) is the normalization of 𝐸 𝑃 . In the function field case, we have
                                       2𝑔(Nor(𝐸 𝑃 )) − 2
                            𝑑 (𝑃) =                           − (2𝑔(𝐵) − 2).
                                              [𝐹 : 𝑘]
So the difference between 𝑑 𝑎 (𝑃) and 𝑑 (𝑃) is related to the difference between the arithmetic
and geometric genera. There are some elementary properties of 𝑑 𝑎 .
Lemma 4.2.3. Let 𝑋, 𝐵, 𝐹 defined as before. The following hold:
                                                                                     ¯ corresponding
    1. If 𝑋 ′ is another model birational to 𝑋, and if 𝑃′ denotes the point in 𝑋 ′ ( 𝑘)
                   ¯ then
        to 𝑃 ∈ 𝑋 ( 𝑘),
                                       𝑑 𝑎 (𝑃′) = 𝑑 𝑎 (𝑃) + 𝑂 ([𝐹 : Q]).
                                                      24


   2. If 𝑋 ′ is the model obtained from 𝑋 by a base change and desingularizing, if 𝑃′ is similarly
      defined, and if the base change is linearly disjoint from 𝐹, then
                                       𝑑 𝑎 (𝑃′) = 𝑑 𝑎 (𝑃) + 𝑂 ([𝐹 : Q]).
   3. If 𝑋 = P1 , then
                                  𝑑 𝑎 (𝑃) = (2[𝐹 : 𝑘] − 2)ℎ(𝑃) + 𝑂 (1).
   4. If 𝑓 : 𝑋 → 𝑌 is a morphism of arithmetic surfaces over 𝐵, and if 𝑓 | 𝐸 𝑃 is generically
      injective, then
                                              𝑑 𝑆 (𝑃) ≤ 𝑑 𝑎 ( 𝑓 (𝑃)).
    A celebrated result on the arithmetic discriminant by Vojta [24], which can be regarded as a
proven weak version of Vojta’s conjecture.
Theorem 4.2.4. Fix an integer 𝜈 ≥ 1, a real number 𝜖 > 0, an effective divisor 𝐷 on 𝑋 with
no multiple components, and a divisor 𝐴 on 𝑋 which is ample on the generic fibre. Then for all
points 𝑃 ∈ 𝐶 (𝑘) \ Supp(𝐷) with [𝑘 (𝑃) : 𝑘] ≤ 𝜈,
                           𝑚(𝐷, 𝑃) + ℎ 𝐾 (𝑃) ≤ 𝑑 𝑎 (𝑃) + 𝜖 ℎ 𝐴 (𝑃) + 𝑂 (1),
where the constant in 𝑂 (1) depends on 𝑋, 𝐷, 𝜈, 𝐴 and 𝜖.
                                                    25


                                             CHAPTER 5
                      TOOLS IN LINEAR RECURRENCE SEQUENCES
Here we give some basic definitions and results involving linear recurrence sequences.
Definition 5.0.1. A linear recurrence is a sequence 𝑎 = (𝑎(𝑖)) of complex numbers satisfying a
homogeneous linear recurrence relation
                    𝑎(𝑖 + 𝑛) = 𝑠1 𝑎(𝑖 + 𝑛 − 1) + · · · + 𝑠𝑛−1 𝑎(𝑖 + 1) + 𝑠𝑛 𝑎(𝑖), 𝑖 ∈ N
with constant coefficients 𝑠 𝑗 ∈ C.
Definition 5.0.2. The polynomial
                               𝑓 (𝑋) = 𝑋 𝑛 − 𝑠1 𝑋 𝑛−1 − · · · − 𝑠𝑛−1 𝑋 − 𝑠𝑛
associated to the relation in Definition 5.0.1 is called its characteristic polynomial and the roots
of this polynomial are said to be its roots.
Definition 5.0.3. A generalized power sum is a finite polynomial-exponential sum
                                               𝑚
                                              ∑︁
                                       𝑎(𝑖) =      𝐴 𝑗 (𝑖)𝛼𝑖𝑗 , 𝑖 ∈ N
                                              𝑗=1
with polynomial coefficients 𝐴 𝑗 (𝑧) ∈ C[𝑧]. The 𝛼 𝑗 are the roots of the sequence 𝑎(𝑖).
    It is a well-known fact that every linear recurrence sequence 𝑎(𝑥) can be written in the
form of a generalized power sum and in fact these two forms are equivalent, see [7]. Through-
out this thesis, linear recurrence sequences are presented in the form of a generalized power sum.
    The linear recurrence sequence 𝑎(𝑖) is called degenerate if it has a pair of distinct roots
whose ratio is a root of unity. Otherwise, it is called non-degenerate.
                                                    26


    Fix a number field 𝑘. Let us define two linear recurrence sequences 𝐹 (𝑛) and 𝐺 (𝑛) by
generalized power sums
                                                            𝑚
                                                           ∑︁
                                              𝐹 (𝑛) =           𝐴𝑖 (𝑛)𝛼𝑖𝑛
                                                           𝑖=1
                                                           ∑︁𝑙
                                             𝐺 (𝑛) =            𝐵𝑖 (𝑛) 𝛽𝑖𝑛
                                                           𝑖=1
where 𝐴(𝑛), 𝐵(𝑛) are polynomials over 𝑘 and 𝛼𝑖 and 𝛽𝑖 are roots in 𝑘 ∗ . Let Γ be the multiplicative
group generated by all 𝛼𝑖 and 𝛽𝑖 with a set of generators {𝑢 1 , . . . , 𝑢𝑟 }. Then we can write 𝐹 (𝑛)
and 𝐺 (𝑛) as
                                         𝐹 (𝑛) = 𝑓 (𝑛, 𝑢 1𝑛 , . . . , 𝑢𝑟𝑛 )
                                         𝐺 (𝑛) = 𝑔(𝑛, 𝑢 1𝑛 , . . . , 𝑢𝑟𝑛 )
where 𝑓 and 𝑔 are rational functions in 𝑥0 , . . . , 𝑥𝑟 of the form:
                                                                𝑓˜(𝑥 0 , . . . , 𝑥𝑟 )
                                      𝑓 (𝑥 0 , . . . , 𝑥𝑟 ) =
                                                                  𝑥 1𝑎1 · · · 𝑥𝑟𝑎𝑟
                                                                ˜ 0 , . . . , 𝑥𝑟 )
                                                               𝑔(𝑥
                                      𝑔(𝑥0 , . . . , 𝑥𝑟 ) =
                                                                  𝑥1𝑏1 · · · 𝑥𝑟𝑏𝑟
with 𝑓˜, 𝑔˜ polynomials, i.e., 𝑓 , 𝑔 ∈ 𝑘 [𝑥 0 , 𝑥0−1 , . . . , 𝑥𝑟 , 𝑥𝑟−1 ] are Laurent polynomials. In particular,
the ring of such Laurent polynomials is a localization of 𝑘 [𝑥0 , . . . , 𝑥𝑟 ], so it is a UFD.
    It is obvious that linear recurrence sequences are closed under term-wise sum and product
from the generalized power sum point of view, hence we can talk about the sum and product of
two recurrence sequences. Let HΓ (𝑘) be the ring of linear recurrence sequences whose coeffi-
cient polynomials are over 𝑘 and roots belonging to a torsion-free multiplicative group Γ ⊂ 𝑘 ∗ .
We say 𝐹 (𝑛), 𝐺 (𝑛) ∈ HΓ (𝑘) are coprime if there does not exist a non-unit 𝐻 (𝑛) ∈ HΓ (𝑘) such
that 𝐹 (𝑛) = 𝐻 (𝑛)𝐹0 (𝑛) and 𝐺 (𝑛) = 𝐻 (𝑛)𝐺 0 (𝑛) with 𝐹0 (𝑛), 𝐺 0 (𝑛) ∈ HΓ (𝑘). Recall that for
𝐹 (𝑛), 𝐺 (𝑛) ∈ HΓ (𝑘) and a choice of generators of the torsion-free group Γ, there are associated
Laurent polynomials 𝑓 and 𝑔 respectively; if two such recurrence sequences are coprime then
                                                            27


the two associated Laurent polynomials are also coprime.
    We also need a well-known theorem on the structure of the zeros of a linear recurrence:
Theorem 5.0.4 (Skolem-Mahler-Lech). The set of indices of the zeros of a linear recurrence
sequence comprises a finite set together with a finite number of arithmetic progressions. If the
linear recurrence sequence is nondegenerate, then there are only finitely many zeros.
                                              28


                                             CHAPTER 6
                  ALMOST 𝑆-UNITS AND ALMOST 𝑆-UNIT EQUATIONS
6.1     Compatible definitions
The definition of almost 𝑆, 𝛿-units was already given as in Definition 1.1.6. Here are some
remarks about this definition and its properties.
Remark 6.1.1. Silverman has defined “quasi-𝑆-integers" in [17]. For a number field 𝑘, a finite
set of places 𝑆 and 𝜖 > 0, the set of quasi-𝑆-integers are defined as
                                                 ∑︁
                            𝑅𝑆 (𝜖) := {𝑥 ∈ 𝑘 :       max{|𝑥| 𝑣 , 0} ≥ 𝜖 ℎ(𝑥)}.
                                                 𝑣∈𝑆
Silverman’s notion of quasi-𝑆-integers can be compared with our notion of almost (𝑆, 𝛿)-units
as follows: if 𝑥 ∈ 𝑘 𝑆,1−𝜖 then 𝑥 ∈ 𝑅𝑆 (𝜖), and if 𝑥 ∈ 𝑅𝑆 (𝜖) then 𝑥 ∈ 𝑘 𝑆,2−𝜖 .
Remark 6.1.2. We note that 𝑘 𝑆,𝛿  𝑛 ⊂ G𝑛 (𝑘)
                                           𝑚     𝑆,𝑛𝛿 and when 𝛿 = 0 we recover 𝑛-tuples of 𝑆-units,
                 ∗ )𝑛.
𝐺 𝑛𝑚 (𝑘) 𝑆,0 = (O𝑘,𝑆
Remark 6.1.3. We use projective height to define almost 𝑆-units in G𝑛𝑚 (𝑘) 𝑆,𝛿 . In other references
standard height is frequently used, where for a point 𝑃 = (𝑥 1 , . . . , 𝑥 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 ,
                                                         𝑛
                                                        ∑︁
                                       ℎ 𝑠𝑡𝑎𝑛𝑑 (𝑃) :=       ℎ(𝑥 𝑛 ).
                                                        𝑖=1
Local heights are defined similarly as
                                                         𝑛
                                                        ∑︁
                                      𝜆 𝑠𝑡𝑎𝑛𝑑,𝑣 (𝑃) :=      𝜆 𝑣 (𝑥 𝑛 ).
                                                        𝑖=1
One can verify that if 𝑃 ∈ G𝑛𝑚 (𝑘) is an (𝑆, 𝛿)-unit under the projective height, then it is an
                                                     29


(𝑆, 𝑛𝛿)-unit under the standard height. Indeed, in this case we have
                        ∑︁                                           ∑︁ ∑︁ 𝑛
                            𝜆 𝑠𝑡𝑎𝑛𝑑,𝑣 (𝑃) + 𝜆 𝑠𝑡𝑎𝑛𝑑,𝑣 (1/𝑃) =           (       𝜆 𝑣 (𝑥𝑖 ) + 𝜆 𝑣 (1/𝑥𝑖 ))
                        𝑣∉𝑆                                          𝑣∉𝑆 𝑖=1
                                                                       ∑︁
                                                                 ≤𝑛         𝜆 𝑣 (𝑃) + 𝜆 𝑣 (1/𝑃)
                                                                       𝑣∉𝑆
                                                                 ≤ 𝑛𝛿ℎ(𝑃) ≤ 𝑛𝛿ℎ 𝑠𝑡𝑎𝑛𝑑 (𝑃).
6.2     Almost 𝑆-unit equation theorem
Before the main proof, we need a generalized version of the unit equation.
Lemma 6.2.1. Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘 containing all
archimedean places. Let 0 < 𝛿 < 1/((𝑛 + 1)(𝑛 + 2)). Let 𝜒 be the set of solutions of
                                                                                     𝑛+1
                                     𝑥 0 + · · · + 𝑥 𝑛 = 1, (𝑥0 , . . . , 𝑥 𝑛 ) ∈ 𝑘 𝑆,𝛿   ,
such that no proper subsum of 𝑥0 + · · · + 𝑥 𝑛 vanishes. Then 𝜒 is a finite set.
Proof. Let (𝑎 0 , . . . , 𝑎 𝑛 ) be a solution in 𝑘 𝑆,𝛿 𝑛+1 and 𝑃 = (𝑎 : . . . : 𝑎 ) ∈ P𝑛 . Let 𝐻 , 𝑖 = 0, . . . , 𝑛
                                                                            0               𝑛            𝑖
be hyperplanes defined by 𝑥𝑖 = 0, 𝐻𝑛+1 be the hyperplane defined by 𝑥 0 + · · · + 𝑥 𝑛 = 0. Let
𝑃′ = (𝑎 0 , . . . , 𝑎 𝑛 ). Note that every coordinate of 𝑃′ is in 𝑘 𝑆,𝛿 , and by easy calculations and
Remark 6.1.2, we know 𝑃′ ∈ G𝑛+1          𝑚 (𝑘) 𝑆,(𝑛+1)𝛿 . By triangle inequalities, for any 𝑣 ∈ 𝑀 𝑘,∞ ,
                                  |1| 𝑣 = |𝑎 0 + · · · + 𝑎 𝑛 | 𝑣 ≤ (𝑛 + 1) max |𝑎𝑖 | 𝑣 ,
                                                                                  𝑖
and for 𝑣 ∉ 𝑀𝑘,∞
                                        |1| 𝑣 = |𝑎 0 + · · · + 𝑎 𝑛 | 𝑣 ≤ max |𝑎𝑖 | 𝑣 .
                                                                            𝑖
It follows that for all 𝑣 ∈ 𝑀𝑘
                             ℎ(𝑃′) = ℎ(𝑃) + 𝑂 (1) and 𝜆 𝑣 (𝑃′) = 𝜆 𝑣 (𝑃) + 𝑂 (1).
                                                             30


Hence we get
                        𝑛+1 ∑︁
                        ∑︁
                                 𝜆 𝐻𝑖 ,𝑣 (𝑃) ≥ (𝑛 + 2 − (𝑛 + 1)(𝑛 + 2)𝛿)ℎ(𝑃) + 𝑂 (1).
                        𝑖=0 𝑣∈𝑆
Applying the Subspace theorem, we have
                        (𝑛 + 2 − (𝑛 + 1)(𝑛 + 2)𝛿)ℎ(𝑃) ≤ (𝑛 + 1 + 𝜖)ℎ(𝑃) + 𝑂 (1)
unless 𝑃 lies in some certain proper linear subspaces of P𝑛 . For a fixed 𝛿 < 1/((𝑛 + 1)(𝑛 + 2)),
taking 𝜖 sufficiently small, the above implies such 𝑃 is contained in a finite union of hyperplanes
in P𝑛 . If 𝑛 = 1, we are done. Otherwise, we proceed by induction as in the proof of the standard
unit equation [2, Theorem 7.4.2].                                                                     □
Corollary 6.2.2. Let 0 < 𝛿 < 1/((𝑛 + 1)(𝑛 + 2)). Let 𝜒 be the set of solutions of
                                                𝑥0 + · · · + 𝑥𝑛 = 1
                                  𝑛+1 . Then there is a finite set F ⊂ 𝑘 ∗ such that every x ∈ 𝜒 has at
such that (𝑥0 , . . . , 𝑥 𝑛 ) ∈ 𝑘 𝑆,𝛿
least one coordinate in F .
Proof. The proof follows from Lemma 6.2.1 and induction.
                                                                                                      □
    Lemma 6.2.1 and Corollary 6.2.2 together give the generalized unit equation for 𝑘 𝑆,𝛿     𝑛 , which
allows us to obtain finiteness of solutions in several of the following theorems.
                                                         31


                                                      CHAPTER 7
              PROOFS OF DIOPHANTINE APPROXIMATION THEOREMS
In this section, our main goal is to give the proof of Theorem 1.1.7.
    In the following we will use the notation u and i for 𝑛-tuples (𝑢 1 , . . . , 𝑢 𝑛 ) and (𝑖1 , . . . , 𝑖 𝑛 ),
respectively, with |i| = 𝑖1 + · · · + 𝑖 𝑛 and denote by ui the multi-variable monomial 𝑢𝑖11 · · · 𝑢𝑖𝑛𝑛 . Let
𝑚 be a positive integer. For a subset 𝑇 ⊂ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ], we let
                                               𝑇𝑚 = {𝑝 ∈ 𝑇 | deg 𝑝 ≤ 𝑚},
and
                                𝑇[𝑚] = {𝑝 ∈ 𝑇 | 𝑝 is homogeneous of degree 𝑚}.
For 𝑓 , 𝑔 ∈ 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ], we let
                                   ( 𝑓 , 𝑔) (𝑚) = { 𝑓 𝑝 + 𝑔𝑞| deg 𝑓 𝑝, deg 𝑔𝑞 ≤ 𝑚},
where deg denotes the (total) degrees of the polynomials.
    Before the proof, we need a combinatorial lemma.
Lemma 7.0.1. Let 𝑚 be a positive integer. Let 𝐼 = {i = (𝑖 0 , . . . , 𝑖 𝑛 )} be the set of (𝑛 + 1)-tuples
in N𝑛+1 with 𝑖0 + · · · + 𝑖 𝑛 = 𝑚. Then
                                                               

                                                ∑︁      𝑚 𝑛+𝑚
                                                            𝑛
                                                    i=           (1, . . . , 1)
                                                i∈𝐼
                                                         𝑛+1
where addition and scalar multiplication are coordinate-wise.
    We also need Lemma 2.1 from [Corvaja et al.].
                                                            32


Lemma 7.0.2. Let 𝐹1 , 𝐹2 ∈ 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ] be coprime homogeneous polynomials of degrees 𝑑1
and 𝑑2 , respectively. Let 𝐵 ⊂ 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ] [𝑚] be a set of monomials of degree 𝑚 whose images
are linearly independent in 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ] [𝑚] /(𝐹1 , 𝐹2 ) [𝑚] . Then
                                                                                                       
             ∑︁
                          j      𝑚+𝑛          𝑚 + 𝑛 − 𝑑1           𝑚 + 𝑛 − 𝑑2             𝑚 + 𝑛 − 𝑑1 − 𝑑2
                   ord𝑥𝑖 x ≤              −                    −                     +
              j
                                 𝑛+1              𝑛+1                  𝑛+1                      𝑛+1
            x ∈𝐵
                                                  
                                       𝑚+𝑛−2
                            ≤ 𝑑1 𝑑2
                                         𝑛−1
for 𝑖 = 0, . . . , 𝑛.
Proof. Let 𝑆 = 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ]. For an 𝑙 ∈ N and a graded module 𝑀 over 𝑆, let 𝑑 𝑀 (𝑙) =
dim 𝑘 𝑀 [𝑙] . Let 𝐼 be an ideal generated by a homogeneous polynomial of degree 𝑖. By the
well-known theory of Hilbert polynomials, 𝑑 𝑆/𝐼 (𝑙) = 𝑑 𝑆 (𝑙) − 𝑑 𝑆 (𝑙 − 𝑖). In this case,
        dim(𝑆 [𝑙] /(𝐹1 , 𝐹2 ) [𝑙] ) = 𝑑 𝑆/(𝐹1 ) (𝑙) − 𝑑 𝑆/(𝐹1 ) (𝑙 − 𝑑2 )
                                    = 𝑑 𝑆 (𝑙) − 𝑑 𝑆 (𝑙 − 𝑑1 ) − (𝑑 𝑆 (𝑙 − 𝑑2 ) − 𝑑 𝑆 (𝑙 − 𝑑1 − 𝑑2 ))
                                                                                                         
                                         𝑙+𝑛          𝑙 + 𝑛 − 𝑑1          𝑙 + 𝑛 − 𝑑2         𝑙 + 𝑛 − 𝑑1 − 𝑑2
                                    =            −                  −                     +                       .
                                           𝑛                𝑛                   𝑛                      𝑛
Let 𝑖 ∈ {0, . . . , 𝑛}, let 𝑆′[𝑙] be the image of 𝑥𝑖𝑙 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ] [𝑚−𝑙] in 𝑆 [𝑚] /(𝐹1 , 𝐹2 ) [𝑚] . Notice that
                        ∑︁                 𝑚
                                          ∑︁                                         𝑚
                                                                                    ∑︁
                                    j
                             ord𝑥𝑖 x ≤        𝑗 (dim 𝑆′[ 𝑗]   − dim 𝑆′[ 𝑗+1] )   =        dim 𝑆′[ 𝑗] ,
                       xj ∈𝐵              𝑗=1                                       𝑗=1
and that dim 𝑆′[𝑙] ≤ dim 𝑆 [𝑚−𝑙] /(𝐹1 , 𝐹2 ) [𝑚−𝑙] , hence we have
                                    ∑︁                 𝑚−1
                                                       ∑︁
                                         ord𝑥𝑖 xj ≤         dim 𝑆 [ 𝑗] /(𝐹1 , 𝐹2 ) [ 𝑗] .
                                   xj ∈𝐵                𝑗=0
Using Pascal’s identity for binomial coefficients,
                                                                                                       
             ∑︁
                          j      𝑚+𝑛          𝑚 + 𝑛 − 𝑑1           𝑚 + 𝑛 − 𝑑2             𝑚 + 𝑛 − 𝑑1 − 𝑑2
                   ord𝑥𝑖 x ≤              −                    −                     +
              j
                                 𝑛+1              𝑛+1                  𝑛+1                      𝑛+1
            x ∈𝐵
                                                  
                                       𝑚+𝑛−2
                            ≤ 𝑑1 𝑑2                  .
                                         𝑛−1
                                                                                                                       □
                                                            33


Theorem 7.0.3. Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘 containing the
archimedean places. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] be coprime polynomials. For all 0 < 𝛿 < 1, there
exists a proper Zariski closed subset 𝑍 of G𝑛𝑚 such that
                       ∑︁                                                                                ∑︁
                 −           log− max{| 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 , |𝑔(𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 } < 𝐶𝛿1/2       ℎ(𝑢𝑖 )
                     𝑣∈𝑀𝑘 \𝑆                                                                            1≤𝑖≤𝑛
for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍, where 𝐶 = 2(𝑛2 deg 𝑓 + 𝑛 deg 𝑔) is a constant.
Proof. This proof is modeled on the proof of Theorem 3.2 of [14].
     Consider the ideal ( 𝑓 , 𝑔) ⊂ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ]. We first assume that ( 𝑓 , 𝑔) (𝑚) ≠ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] 𝑚 .
It follows that the 𝑘-vector space 𝑉𝑚 = 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] 𝑚 /( 𝑓 , 𝑔) (𝑚) is not trivial. Let u =
(𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 . For 𝑣 ∈ 𝑆, we construct a basis 𝐵𝑣 for 𝑉𝑚 as follows. Choose a mono-
mial xi1 ∈ 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] 𝑚 so that |ui1 | 𝑣 is minimal subject to the condition xi1 ∉ ( 𝑓 , 𝑔) (𝑚) . Sup-
pose now that xi1 , . . . , xi 𝑗 have been constructed and are linearly independent modulo ( 𝑓 , 𝑔) (𝑚) ,
but don’t span 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] 𝑚 modulo ( 𝑓 , 𝑔) (𝑚) . Then we let xi 𝑗+1 ∈ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] 𝑚 be a
monomial such that |ui 𝑗+1 | 𝑣 is minimal subject to the condition that xi1 , . . . , xi 𝑗+1 are linearly
independent modulo ( 𝑓 , 𝑔) (𝑚) . In this way, we construct a basis of 𝑉𝑚 with monomial represen-
tatives xi1 , . . . , xi 𝑁 ′ , where 𝑁 ′ = 𝑁𝑚′ = dim 𝑉𝑚 . Let 𝐼𝑣 = {i1 , . . . , i𝑁 ′ }. We also choose a basis
𝜙1 , . . . , 𝜙 𝑁 of the vector space ( 𝑓 , 𝑔) (𝑚) , where 𝑁 = 𝑁𝑚 = dim( 𝑓 , 𝑔) (𝑚) . Now for i, |i| ≤ 𝑚, we
have that
                                                      ∑︁𝑁′
                                                 i
                                                x +          𝑐 i, 𝑗 xi 𝑗 ∈ ( 𝑓 , 𝑔) (𝑚)
                                                       𝑗=1
for some choice of coefficients 𝑐 i, 𝑗 ∈ 𝑘. Then for each such i there is a linear form 𝐿 i𝑣 over 𝑘
such that
                                                                              ∑︁𝑁′
                                                                          i
                                          𝐿 i𝑣 (𝜙1 , . . . , 𝜙 𝑁 )   =x +           𝑐 i, 𝑗 xi 𝑗 .
                                                                               𝑗=1
                                                                    34


Note that {𝐿 i𝑣 (𝜙1 , . . . , 𝜙 𝑁 ) : |i| ≤ 𝑚, i ∉ 𝐼𝑣 } is a basis for ( 𝑓 , 𝑔) (𝑚) , and {𝐿 i𝑣 : |i| ≤ 𝑚, i ∉ 𝐼𝑣 } is
a set of 𝑁 linearly independent forms in 𝑁 variables. Let
                                     𝑃 = 𝜙(u) := (𝜙1 (u), . . . , 𝜙 𝑁 (u)) ∈ 𝑘 𝑁 .
We may additionally assume that 𝜙(u) ≠ 0 (by enlarging the set 𝑍). From the triangle inequality
and the definition of xi1 , . . . , xi 𝑁 ′ , for any i with |i| ≤ 𝑚, i ∉ 𝐼𝑣 , we have the key inequality
                                            log |𝐿 i𝑣 (𝑃)| 𝑣 ≤ log |ui | 𝑣 + 𝐶𝑣
where the constant 𝐶𝑣 depends only on 𝑣 ∈ 𝑆 and the set {i1 , . . . , i𝑁 ′ } (and not on u).
     We will apply the Subspace Theorem with the choice of linear forms 𝐿 i𝑣 , |i| ≤ 𝑚, i ∉ 𝐼𝑣 , for
each 𝑣 ∈ 𝑆. We want to estimate the sum
                                               ∑︁ ∑︁                    |𝑃| 𝑣
                                                               log                .
                                               𝑣∈𝑆 |i|≤𝑚,i∉𝐼 𝑣
                                                                    |𝐿 i (𝑃)| 𝑣
                                                                        𝑣
Towards this end, we estimate the sums
                              ∑︁ ∑︁                                       ∑︁       ∑︁
                           −                   log |𝐿 i𝑣 (𝑃)| 𝑣 and                       log |𝑃| 𝑣
                               𝑣∈𝑆 |i|≤𝑚,i∉𝐼 𝑣                            𝑣∈𝑆 |i|≤𝑚,i∉𝐼 𝑣
separately.
     We have
                          ∑︁       ∑︁                              ∑︁       ∑︁
                      −                   log |𝐿 i𝑣 (𝑃)| 𝑣 ≥ −                       log |ui | 𝑣 − 𝐶𝑁
                          𝑣∈𝑆 |i|≤𝑚,i∉I𝑣                           𝑣∈𝑆 |i|≤𝑚,i∉I𝑣
             ∑︁
where 𝐶 =        𝐶𝑣 . By the product formula,
             𝑣∈𝑆
                              ∑︁                   ∑︁                      ∑︁
                                   log |ui | 𝑣 +           log |ui | 𝑣 =          log |ui | 𝑣 = 0.
                              𝑣∈𝑆                𝑣∈𝑀𝑘 \𝑆                  𝑣∈𝑀𝑘
It follows that,
                     ∑︁         ∑︁                        ∑︁ ∑︁                      ∑︁ ∑︁
                  −                    log |ui | 𝑣 = −                log |ui | 𝑣 +            log |ui | 𝑣
                      𝑣∈𝑆 |i|≤𝑚,i∉I𝑣                      𝑣∈𝑆 |i|≤𝑚                  𝑣∈𝑆 i∈I𝑣
                                                       ∑︁ ∑︁                        ∑︁ ∑︁
                                                   =             log |ui | 𝑣 +                   log |ui | 𝑣 .
                                                       𝑣∈𝑆 i∈I𝑣                  𝑣∈𝑀𝑘 \𝑆 |i|≤𝑚
                                                               35


   Let 𝑑1 = deg 𝑓 and 𝑑2 = deg 𝑔. By Lemma 7.0.2, we have
                                                                           
                               ∑︁ ∑︁
                                                i               𝑚 + 𝑛 − 2 ∑︁
                            −            log |u | 𝑣 ≤ 𝑑1 𝑑2                             ℎ(𝑢𝑖 ),
                               𝑣∈𝑆 i∈I
                                                                   𝑛−1        1≤𝑖≤𝑛
                                       𝑣
we find that,
                                                                               
                    ∑︁      ∑︁                                      𝑚 + 𝑛 − 2 ∑︁
                 −                  log |𝐿 i𝑣 (𝑃)| 𝑣  ≥ − 𝑑1 𝑑2                           ℎ(𝑢𝑖 ) − 𝐶𝑁
                    𝑣∈𝑆 |i|≤𝑚,i∉I𝑣
                                                                      𝑛−1         1≤𝑖≤𝑛
                                                             ∑︁ ∑︁
                                                        +                 log |ui | 𝑣 .
                                                           𝑣∈𝑀𝑘 \𝑆 |i|≤𝑚
By Lemma 7.0.1,
                           ∑︁ ∑︁                        ∑︁ ∑︁
                                         log |ui | 𝑣 =                log |ui | 𝑣
                         𝑣∈𝑀𝑘 \𝑆 |i|≤𝑚                 |i|≤𝑚 𝑣∈𝑀𝑘 \𝑆
                                                                

                                                        𝑚 𝑛+𝑚𝑛
                                                                     ∑︁     ∑︁
                                                     =                            log |𝑢𝑖 | 𝑣
                                                          𝑛+1
                                                                   𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
                                                              𝑛+𝑚 
 ∑︁                        
                                                          𝑚     𝑛
                                                                              ∑︁            1
                                                     ≥−                               𝜆𝑣         .
                                                            𝑛+1                             𝑢𝑖
                                                                     𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
So we estimate,
                                                                ∑︁                     𝑛+𝑚 
 ∑︁                    
     ∑︁      ∑︁                                     𝑚 +  𝑛 −  2                     𝑚     𝑛
                                                                                                         ∑︁        1
  −                 log |𝐿 i𝑣 (𝑃)| 𝑣 ≥ −𝑑1 𝑑2                            ℎ(𝑢𝑖 ) −                             𝜆𝑣
                                                      𝑛−1                              𝑛 +  1                      𝑢𝑖
     𝑣∈𝑆 |i|≤𝑚,i∉I𝑣                                               1≤𝑖≤𝑛                         𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
                                     − 𝐶𝑁.
   On the other hand,
                 ∑︁      ∑︁                        ∑︁                                   ∑︁
                                log |𝑃| 𝑣 = 𝑁          log |𝑃| 𝑣 = 𝑁 (ℎ(𝑃) −                  log |𝑃| 𝑣 ).
                 𝑣∈𝑆 |i|≤𝑚,i∉I𝑣                    𝑣∈𝑆                              𝑣∈𝑀𝑘 \𝑆
   Now since 𝜙𝑖 = 𝑓 𝑝𝑖 + 𝑔𝑞𝑖 , deg 𝑓 𝑝𝑖 , deg 𝑔𝑞𝑖 ≤ 𝑚, we have for 𝑣 ∈ 𝑀𝑘 \ 𝑆,
                  log |𝜙𝑖 (u)| 𝑣 = log | 𝑓 𝑝𝑖 (u) + 𝑔𝑞𝑖 (u)| 𝑣
                                 ≤ log max{| 𝑓 𝑝𝑖 (u)| 𝑣 , |𝑔𝑞𝑖 (u)| 𝑣 } + 𝑂 𝑣 (1)
                                 ≤ log− max{| 𝑓 𝑝𝑖 (u)| 𝑣 , |𝑔𝑞𝑖 (u)| 𝑣 } + 𝑚𝜆 𝑣 (u) + 𝑂 𝑣 (1)
                                 ≤ log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } + 𝑚𝜆 𝑣 (u) + 𝑂 𝑣 (1),
                                                           36


where 𝑂 𝑣 (1) = 0 for all but finitely many 𝑣.
    Then for 𝑣 ∈ 𝑀𝑘 \ 𝑆,
                         log |𝑃| 𝑣 ≤ log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } + 𝑚𝜆 𝑣 (u) + 𝐶𝑣 .
    Now we sum over 𝑣 ∈ 𝑀𝑘 \ 𝑆 to get:
             ∑︁                    ∑︁                                                ∑︁
                   log |𝑃| 𝑣 ≤            log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } + 𝑚              𝜆 𝑣 (u) + 𝑂 (1).
           𝑣∈𝑀𝑘 \𝑆               𝑣∈𝑀𝑘 \𝑆                                           𝑣∈𝑀𝑘 \𝑆
    Then we find the estimate:
                 ∑︁ ∑︁                                       ∑︁
                                log |𝑃| 𝑣 ≥𝑁 (ℎ(𝑃) −               log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 }
                 𝑣∈𝑆 |i|≤𝑚,i∉I𝑣                            𝑣∈𝑀𝑘 \𝑆
                                                     ∑︁      ∑︁
                                              −𝑚                   𝜆 𝑣 (𝑢𝑖 )) + 𝑂 (1).
                                                   𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
    One also has the easy estimate
                                             ℎ(𝑃) ≤ 𝑚ℎ(u) + 𝑂 (1).
    Schmidt’s Subspace Theorem implies that there exists a finite union 𝑍 of proper subspaces
of 𝑘 𝑁 such that
                                 ∑︁       ∑︁            |𝑄| 𝑣
                                                 log             ≤ (𝑁 + 1)ℎ(𝑄)
                                  𝑣∈𝑆 |i|≤𝑚,i∉I𝑣
                                                     |𝐿 i (𝑄)| 𝑣
                                                        𝑣
for all 𝑄 ∈ 𝑘 𝑁 \ 𝑍.
    Using the above estimates, if 𝑃 = 𝜙(u) ∉ 𝑍, we find that up to an 𝑂 (1),
                      ∑︁                                                ∑︁ ∑︁
      𝑁 ­ ℎ(𝑃) −            log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } − 𝑚                      𝜆 𝑣 (𝑢𝑖 ) ®
         ©                                                                                      ª
         «          𝑣∈𝑀𝑘 \𝑆                                           𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛             ¬
                             ∑︁                   𝑛+𝑚 
 ∑︁                    
                 𝑚+𝑛−2                           𝑚   𝑛
                                                                    ∑︁          1
       − 𝑑1 𝑑2                        ℎ(𝑢𝑖 ) −                             𝜆𝑣        ≤ (𝑁 + 1)ℎ(𝑃) + 𝐶𝑁.
                    𝑛−1        1≤𝑖≤𝑛
                                                  𝑛+1              1≤𝑖≤𝑛
                                                                                𝑢𝑖
                                                           𝑣∈𝑀𝑘 \𝑆
                                                          37


Applying the estimate for ℎ(𝑃), combining terms, and dividing by 𝑁, we obtain up to an 𝑂 (1),
                                                                                                    

                                                                                          𝑚 𝑛+𝑚
                                                                                                                           
       ∑︁
                −
                                                         ∑︁ ∑︁
                                                                                                 𝑛
                                                                                                        ∑︁ ∑︁              1
  −         log max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } − 𝑚                          𝜆 𝑣 (𝑢𝑖 ) −                                   𝜆𝑣
                                                                                         𝑁 (𝑛 + 1)                         𝑢𝑖
    𝑣∈𝑀𝑘 \𝑆                                            𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛                                  𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
                          

     𝑚 + 𝑑1 𝑑2 𝑚+𝑛−2
                   𝑛−1
                               ∑︁
  ≤                                ℎ(𝑢𝑖 ).
              𝑁              1≤𝑖≤𝑛
    Since 𝑓 and 𝑔 are coprime, the ideal ( 𝑓 , 𝑔) defines a closed subset of A𝑛 of codimension
at least 2. Without loss of generality, assume 𝑑1 ≥ 𝑑2 . By Lemma 7.0.2, we find that
                                                         1 −𝑑 2
𝑁 ′ = 𝑚+𝑛                                                                                                               − 𝑁′ ≥
            
                
               
                  
                         
                           

         𝑛     − 𝑚+𝑛−𝑑𝑛
                           1
                               − ( 𝑚+𝑛−𝑑
                                       𝑛
                                           2
                                               − 𝑚+𝑛−𝑑 𝑛          ) ≤ 𝑑1 𝑑2 𝑚+𝑛−2  𝑛−2       and that 𝑁 = 𝑚+𝑛     𝑛
 𝑚+𝑛 
                  

  𝑛    − 𝑑1 𝑑2 𝑚+𝑛−2
                  𝑛−2 . We assume now 𝑚 ≥ 𝑑 1 𝑛. Then we have the estimate
                                                               
                    𝑚+𝑛                   𝑚+𝑛−2               𝑚+𝑛                        𝑑1 𝑑2 𝑛(𝑛 − 1)
                                − 𝑑1 𝑑2                                  =1−
                       𝑛                     𝑛−2                  𝑛                 (𝑚 + 𝑛)(𝑚 + 𝑛 − 1)
                                                                                    𝑑1 𝑑2 𝑛(𝑛 − 1)
                                                                         ≥ 1−
                                                                                            𝑑12 𝑛2
                                                                                    𝑛−1 1
                                                                         ≥ 1−                  = .
                                                                                       𝑛          𝑛
Therefore we have
                                                                                𝑚+𝑛−2
 ∑︁
                ∑︁                                               𝑚  +  𝑑 1 𝑑 2
           −          log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤                     𝑚+𝑛 

                                                                                  𝑛−1
                                                                                                    ℎ(𝑢𝑖 )
              𝑣∈𝑀𝑘 \𝑆
                                                                      1/𝑛      𝑛            1≤𝑖≤𝑛
                                                                     ∑︁ ∑︁
                                                            +𝑚                        𝜆 𝑣 (𝑢𝑖 )
                                                                   𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
                                                                         

                                                                𝑚 𝑚+𝑛 𝑛 /(𝑛 + 1)
                                                                                                                 
                                                                                            ∑︁ ∑︁              1
                                                            +                    
                        𝜆𝑣        .
                                                                    1/𝑛 𝑚+𝑛  𝑛           𝑣∈𝑀𝑘 \𝑆 1≤𝑖≤𝑛
                                                                                                               𝑢𝑖
One shall notice that
                                                       𝑚+𝑛−2

                                          𝑚 + 𝑑1 𝑑2      𝑛−1           2𝑑1 𝑑2 𝑛2
                                                     𝑚+𝑛 
         ≤                ,
                                                1/𝑛   𝑛
                                                                         𝑚+1
and that                                                      

                                                    𝑚 𝑚+𝑛 𝑛
                                                      𝑛 + 1 
 ≤ 𝑚.
                                                   1/𝑛 𝑚+𝑛  𝑛
                                                   ∑︁                        ∑︁
By Remark 6.1.3, hence the condition                    ℎ 𝑆¯ (𝑢𝑖 ) ≤ 𝑛𝛿              ℎ(𝑢𝑖 ) is satisfied, we get
                                                  1≤𝑖≤𝑛                     1≤𝑖≤𝑛
                                                                      2𝑑1 𝑑2 𝑛2
                  ∑︁                                                                           ∑︁
                                −
              −           log max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤                         + 𝑚𝑛𝛿              ℎ(𝑢𝑖 ).
                                                                        𝑚+1                      1≤𝑖≤𝑛
                𝑣∈𝑀𝑘 \𝑆
                                                            38


                                      
                                 2𝑑1 𝑛
     Now letting 𝑚 =                     , it follows that
                                 𝛿1/2
                         ∑︁                                                                     ∑︁
                     −          log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤ 2(𝑑1 𝑛2 + 𝑑2 𝑛)𝛿1/2                 ℎ(𝑢𝑖 ).
                       𝑣∈𝑀𝑘 \𝑆                                                               1≤𝑖≤𝑛
     We can see this choice of 𝑚 satisfies the conditions 𝑚 ≥ 𝑑1 𝑛 and 𝑚 ≥ max{𝑑1 , 𝑑2 }. Now
letting 𝐶 (𝑛, 𝑑1 , 𝑑2 ) = 2(𝑑1 𝑛2 + 𝑑2 𝑛), we have
                          ∑︁                                                                  ∑︁
                      −           log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤ 𝐶 (𝑛, 𝑑1 , 𝑑2 )𝛿1/2              ℎ(𝑢𝑖 )
                        𝑣∈𝑀𝑘 \𝑆                                                             1≤𝑖≤𝑛
as long as u does not lie in the proper closed subset coming from the exceptional set in the
application of the Subspace Theorem.
     Finally, we note that the choice of linear forms in the application of Schmidt’s Subspace
Theorem depends not on u, but on the choice of the monomial bases 𝐵𝑣 , 𝑣 ∈ 𝑆. Since for fixed 𝑚
there are only finitely many monomials of degree at most 𝑚, and hence only finitely many choices
for these bases, we see that for fixed 𝑚 the given argument leads to only finitely many applications
of Schmidt’s Subspace Theorem (over all choices of u). Therefore there exists a proper Zariski
closed subset 𝑍 of G𝑛𝑚 such that the inequality is valid for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍.
     Now consider the case when ( 𝑓 , 𝑔) (𝑚) = 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] 𝑚 . We can find polynomials 𝑓˜, 𝑔˜ ∈
𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] such that
                                                        𝑓 𝑓˜ + 𝑔 𝑔˜ = 1
with deg 𝑓˜, deg 𝑔˜ ≤ 𝑚. Hence for any 𝑣 ∈ 𝑀𝑘 and u ∈ G𝑛𝑚 (𝑠) 𝑆,𝛿 , we have
                     1 = |( 𝑓 𝑓˜ + 𝑔 𝑔)(u)|
                                     ˜                              ˜
                                              𝑣 ≤ max{| 𝑓 (u)| 𝑣 | 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 | 𝑔(u)|
                                                                                         ˜     𝑣}
                                                ≤ max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } max{| 𝑓˜(u)| 𝑣 , | 𝑔(u)|
                                                                                                    ˜     𝑣 }.
Then we have
                               max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≥ min{|1/ 𝑓˜(u)| 𝑣 , |1/𝑔(u)| ˜     𝑣 }.
                                                              39


Applying − log− on both sides and summing over 𝑣 ∈ 𝑀𝑘 \ 𝑆, it follows that
           ∑︁                                                      ∑︁
      −          log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤ −                      log− min{|1/ 𝑓˜(u)| 𝑣 , |1/𝑔(u)|
                                                                                                        ˜      𝑣}
         𝑣∈𝑀𝑘 \𝑆                                                𝑣∈𝑀𝑘 \𝑆
                                                                   ∑︁
                                                         =−                min{log |1/ 𝑓˜(u)| 𝑣 , log |1/𝑔(u)|
                                                                                                           ˜     𝑣 , 0}
                                                                𝑣∈𝑀𝑘 \𝑆
                                                                ∑︁
                                                         =              max{log | 𝑓˜(u)| 𝑣 , log | 𝑔(u)|
                                                                                                   ˜     𝑣 , 0}.
                                                             𝑣∈𝑀𝑘 \𝑆
                                                              ∑︁                      ∑︁
Now since deg 𝑓˜, deg 𝑔˜ ≤ 𝑚, together with                          ℎ 𝑆¯ (𝑢𝑖 ) ≤ 𝛿        ℎ(𝑢𝑖 ) (by Remark 6.1.3), we
                                                            1≤𝑖≤𝑛                    1≤𝑖≤𝑛
obtain
                                ∑︁                                                       ∑︁
                           −          log− max{| 𝑓 (u)| 𝑣 , |𝑔(u)| 𝑣 } ≤ 𝑚𝑛𝛿                  ℎ(𝑢𝑖 ),
                              𝑣∈𝑀𝑘 \𝑆                                                   1≤𝑖≤𝑛
which is an even better estimate according to the proof of the first case.                                              □
                                       𝜖2
    By letting 𝛿 =                                         , we obtain an immediate result:
                         4𝑛2 (𝑛2 deg 𝑓 + 𝑛 deg 𝑔) 2
Corollary 7.0.4. Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘 containing the
archimedean places. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] be coprime polynomials. For all 𝜖 > 0, there
exist 𝛿 > 0 and a proper Zariski closed subset 𝑍 of G𝑛𝑚 such that
            ∑︁
        −         log− max{| 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 , |𝑔(𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 } < 𝜖 max{ℎ(𝑢 1 ), . . . , ℎ(𝑢 𝑛 )}
          𝑣∈𝑀𝑘 \𝑆
for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍.
    The next theorem allows us to control the 𝑆-part of the greatest common divisor in Theorem
1.1.7.
Theorem 7.0.5. Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘 containing the
archimedean places. Let 𝑓 ∈ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] be a polynomial of degree 𝑑 that doesn’t vanish at
the origin (0, . . . , 0). For all 0 < 𝛿 < 1, there exists a proper Zariski closed subset 𝑍 of G𝑛𝑚
such that
                                   ∑︁                                                ∑︁
                                 −      log− | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 < 4𝑛𝑑𝛿           ℎ(𝑢𝑖 )
                                   𝑣∈𝑆                                              1≤𝑖≤𝑛
                                                                40


for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍.
Proof. In the following proof we will not consider the points {(𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 :
 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 ) = 0}. Since this set can be covered by a proper Zariski closed subset, by taking it
into the exceptional set, we can ignore such points.
     For a subset 𝑆′ of 𝑆, let 𝑅𝑆 ′ consist of the set of points (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 such that
                                         𝑆′ = {𝑣 ∈ 𝑆 : log | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 < 0}.
Then for (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ 𝑅𝑆 ′ ,
                                                             
                                                                                                   𝑣 ∈ 𝑆′,
                                                             
                                                              log | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 ,
                                                             
                                                             
                              −
                                                             
                           log | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 =
                                                                                                   𝑣 ∈ 𝑆 \ 𝑆′ .
                                                             
                                                              0,
                                                             
                                                             
                                                             
                                                                                                                𝑛+𝑚𝑑 

     Let 𝑑 = deg 𝑓 , 𝑚 ∈ N and 𝜙 : P𝑛 → P𝑁 , 𝜙 = (𝜙0 , . . . , 𝜙 𝑁 ), 𝑁 =                                         𝑛    − 1, be the
𝑚𝑑-uple embedding of P𝑛 given by the set of monomials of degree 𝑚𝑑 in 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ]. Let
𝐹 = 𝑥0𝑑 𝑓 (𝑥 1 /𝑥 0 , . . . , 𝑥 𝑛 /𝑥 0 ) be the homogenization of 𝑓 in 𝑘 [𝑥0 , . . . , 𝑥 𝑛 ]. Let 𝑉𝑚𝑑 be the vector
space of homogeneous polynomials of degree 𝑚𝑑, and let Mon𝑚𝑑 consist of the set of all mono-
mials in 𝑘 [𝑥 0 , . . . , 𝑥 𝑛 ] of degree 𝑚𝑑.
                                                                                           ord𝑥0 xi                        xi
                                                                                                    
     If 𝑣 ∈      𝑆′, we construct a basis for 𝑉𝑚𝑑          as follows. Let 𝑘 i =                       and define 𝐵i𝑣 = 𝑘 𝑑 𝐹 𝑘 i .
                                                                                                 𝑑                        𝑥0 i
Let 𝐵𝑣 be the set of all 𝐵i𝑣 . Since 𝑓 doesn’t vanish at the origin, 𝑥 0𝑑 appears with a nonzero
coefficient in 𝐹, and thus it’s clear that 𝐵𝑣 is a basis for 𝑉𝑚𝑑 .
     If 𝑣 ∈ 𝑆 \ 𝑆′, then we let 𝐵𝑣 = Mon𝑚𝑑 . Applying the Subspace Theorem on P𝑁 with
appropriate linear forms, we find that for a fixed 𝜖 > 0
                                     ∑︁ ∑︁               |𝜙(𝑃)| 𝑣
                                                  log              ≤ (𝑁 + 1 + 𝜖)ℎ(𝜙(𝑃))
                                     𝑣∈𝑆 𝑄∈𝐵 𝑣
                                                         |𝑄(𝑃)| 𝑣
                                                                  41


for all 𝑃 ∈ P𝑛 (𝑘) \ 𝑍, where 𝑍 = 𝜙−1 (𝑍 ′) and 𝑍 ′ is a finite union of hyperplanes in P𝑁 . From
the definition of 𝐵𝑣 , we can rewrite the left-hand side of above as
              ∑︁        ∑︁             |𝜙(𝑃)| 𝑣 ∑︁ ∑︁                  |𝐵i (𝑃)| 𝑣
                                log                  −           log i𝑣              ≤ (𝑁 + 1 + 𝜖)ℎ(𝜙(𝑃)).
              𝑣∈𝑆 𝑄∈Mon𝑚𝑑
                                       |𝑄(𝑃)| 𝑣          i 𝑣∈𝑆 ′        |x  (𝑃)| 𝑣
Suppose now that (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ 𝑅𝑆 ′ and let 𝑃 = [1 : 𝑢 1 : . . . : 𝑢 𝑛 ] ∈ P𝑛 (𝑘). It follows
immediately that for 𝐵i𝑣 with 𝑘 i 𝑑 ≤ ord𝑥0 xi < (𝑘 i + 1)𝑑,
                                 ∑︁           |𝐵i𝑣 (𝑃)| 𝑣           ∑︁
                              −         log                =  −𝑘  i       log | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 .
                                 𝑣∈𝑆 ′
                                               |xi (𝑃)| 𝑣           𝑣∈𝑆 ′
              Í
Letting 𝐼 =      i 𝑘 i,
                              ∑︁ ∑︁              |𝐵i𝑣 (𝑃)| 𝑣         ∑︁
                          −               log i              = −𝐼          log | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣
                               i   𝑣∈𝑆 ′
                                                  |x (𝑃)| 𝑣          𝑣∈𝑆 ′
                                                                     ∑︁
                                                             = −𝐼          log− | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 .
                                                                     𝑣∈𝑆
By an easy calculation, we find that
                                                                                                             
                            𝑛+𝑑                                         𝑛 + (𝑚 − 1)𝑑                 𝑛 + (𝑚 − 2)𝑑
        𝐼 =1·𝑚+                        − 1 (𝑚 − 1) + · · · +                                   −                           ·1
                              𝑛                                                 𝑛                             𝑛
                                                           
                     𝑛+𝑑                     𝑛 + (𝑚 − 1)𝑑
          =1+                 +···+
                        𝑛                             𝑛
                                  
                𝑛 + (𝑚 − 1)𝑑
          ≥                          .
                        𝑛
    Note that 𝜙 induces a natural map G𝑛𝑚 → G𝑚                    𝑁 and 𝜙(G𝑛 (𝑘) ) ⊂ G 𝑁 (𝑘) . Indeed,
                                                                                  𝑚       𝑆,𝛿           𝑚      𝑆,𝛿
            ∑︁                              ∑︁                                                   ∑︁
                    log |𝜙(𝑃)| 𝑣 =                   log max {|𝑄(𝑃)| 𝑣 } ≤ 𝑚𝑑                              log max |𝑢𝑖 | 𝑣 .
                                                         𝑄∈Mon𝑚𝑑                                                 𝑖
          𝑣∈𝑀𝑘 \𝑆                         𝑣∈𝑀𝑘 \𝑆                                              𝑣∈𝑀𝑘 \𝑆
Similarly, we have
                                   ∑︁                 1                  ∑︁                    1
                                            log              ≤ 𝑚𝑑              log max                .
                                                    𝜙(𝑃)   𝑣                            𝑖     𝑢𝑖 𝑣
                                 𝑣∈𝑀𝑘 \𝑆                              𝑣∈𝑀𝑘 \𝑆
Thus we have
                                                                                  
      ∑︁                                1             ∑︁                              1
            𝜆 𝑣 (𝜙(𝑃)) + 𝜆 𝑣                    ≤            𝑚𝑑 𝜆 𝑣 (𝑃) + 𝜆 𝑣                  ≤ 𝑚𝑑𝛿ℎ(𝑃) ≤ 𝛿ℎ(𝜙(𝑃)).
                                    𝜙(𝑃)                                              𝑃
    𝑣∈𝑀𝑘 \𝑆                                         𝑣∈𝑀𝑘 \𝑆
                                                                 42


Now since 𝜙(𝑃) ∈ G𝑚           𝑁 (𝑘)
                                    𝑆,𝛿 and min |𝜙𝑖 (𝑃)| 𝑣 ≤ |𝑄(𝑃)| 𝑣 for all 𝑄 ∈ Mon𝑚𝑑 , then
                                                    𝑖
                 ∑︁        ∑︁             |𝜙(𝑃)| 𝑣
                                  log
                 𝑣∈𝑆 𝑄∈𝑀𝑜𝑛𝑚𝑑
                                         |𝑄(𝑃)| 𝑣
                      ∑︁       ∑︁               |𝜙(𝑃)| 𝑣        ∑︁         ∑︁                |𝜙(𝑃)| 𝑣
                 =                       log                −                         log
                                                |𝑄(𝑃)| 𝑣                                     |𝑄(𝑃)| 𝑣
                     𝑣∈𝑀𝑘 𝑄∈𝑀𝑜𝑛𝑚𝑑                             𝑣∈𝑀𝑘 \𝑆 𝑄∈𝑀𝑜𝑛𝑚𝑑
                                                     ∑︁       ∑︁             |𝜙(𝑃)| 𝑣
                 = (𝑁 + 1)ℎ(𝜙(𝑃)) −                                    log
                                                                             |𝑄(𝑃)| 𝑣
                                                  𝑣∈𝑀𝑘 \𝑆   𝑄∈𝑀𝑜𝑛 𝑚𝑑
                                                             © ∑︁                               ∑︁          1
                 ≥ (𝑁 + 1)ℎ(𝜙(𝑃)) − (𝑁 + 1) ­                           log |𝜙(𝑃)| 𝑣 +                log
                                                                                                                 ª
                                                                                                                 ®
                                                                                                          𝜙(𝑃) 𝑣
                                                             «𝑣∈𝑀𝑘 \𝑆                         𝑣∈𝑀𝑘 \𝑆            ¬
                 ≥ (𝑁 + 1)(1 − 𝛿)ℎ(𝜙(𝑃)).
Therefore, we have
                                                        ∑︁
            (𝑁 + 1)(1 − 𝛿)ℎ(𝜙(𝑃)) − 𝐼                       log− | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 ≤ (𝑁 + 1 + 𝜖)ℎ(𝜙(𝑃))
                                                        𝑣∈𝑆
for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ 𝑅𝑆 ′ outside of some proper Zariski closed subset 𝑍. It follows that for a
sufficiently small 𝜖
                   ∑︁                                        (𝑁 + 1 + 𝜖/𝛿)
                −        log− | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 <                       𝛿ℎ(𝜙(𝑃))
                   𝑣∈𝑆
                                                                     𝐼
                                                                𝑛+𝑚𝑑 

                                                                  𝑛
                                                          <   𝑛+(𝑚−1)𝑑 

                                                                           𝛿𝑚𝑑ℎ(𝑃)
                                                                  𝑛
                                                             (𝑛 + (𝑚 − 1)𝑑 + 1) · · · (𝑛 + 𝑚𝑑)
                                                          =                                              𝛿𝑚𝑑ℎ(𝑃)
                                                                 ((𝑚 − 1)𝑑 + 1) · · · (𝑚𝑑)
for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ 𝑅𝑆 ′ outside of some proper Zariski closed subset 𝑍. Choose 𝑚 =
  𝑛 − 21/𝑑 + 1
                        
                    + 1 , we have
  𝑑 (21/𝑑 − 1)
                                                    𝑛 + (𝑚 − 1)𝑑 + 1
                                                                            ≤ 21/𝑑 ,
                                                        (𝑚 − 1)𝑑 + 1
        (𝑛 + (𝑚 − 1)𝑑 + 1) · · · (𝑛 + 𝑚𝑑)
hence                                                     ≤ 2. Also notice that 𝑚 ≤ 2𝑛, we obtain
              ((𝑚 − 1)𝑑 + 1) · · · (𝑚𝑑)
                 ∑︁                                                                                       ∑︁
              −         log− | 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 < 2𝛿𝑚𝑑ℎ(𝑃) < 4𝑛𝑑𝛿ℎ(𝑃) < 4𝑛𝑑𝛿                          ℎ(𝑢𝑖 )
                  𝑣∈𝑆                                                                                    1≤𝑖≤𝑛
                                                                  43


for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ 𝑅𝑆 ′ outside of some proper Zariski closed subset 𝑍. In fact, since there
are only finitely many choices of the subset 𝑆′ ⊂ 𝑆, we find that the inequality holds for all
𝑃 ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍, for some proper closed subset 𝑍.
                                                                                                                   □
    From Theorem 7.0.5, the immediate result combined with Theorem 7.0.3 is
Theorem 7.0.6. Let 𝑘 be a number field and let 𝑆 be a finite set of places of 𝑘 containing the
archimedean places. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑥 1 , . . . , 𝑥 𝑛 ] be polynomials that don’t both vanish at the origin
(0, . . . , 0). For all 0 < 𝛿 < 1, there exists a proper Zariski closed subset 𝑍 of G𝑛𝑚 such that
                    ∑︁                                                                                  ∑︁
                −          log− max{| 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 , |𝑔(𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 } < 𝐶𝛿1/2       ℎ(𝑢𝑖 )
                   𝑣∈𝑀𝑘                                                                               1≤𝑖≤𝑛
for all u = (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍, where 𝐶 = 6(deg 𝑓 + deg 𝑔)𝑛2 is a constant.
Proof. With not loss of generality, assume deg 𝑓 ≤ deg 𝑔 and 𝑔 doesn’t vanish at the origin.
Then applying Theorem 7.0.5 to 𝑔, on the right hand side we obtain
                                          ∑︁                                                  ∑︁
                             4𝑛𝛿 deg 𝑔         ℎ(𝑢𝑖 ) < 4𝑛(deg 𝑔 + deg 𝑓 )𝛿                        ℎ(𝑢𝑖 ).
                                         1≤𝑖≤𝑛                                               1≤𝑖≤𝑛
Combining with the inequality from Theorem 7.0.3 finishes the proof.                                               □
    Now we are ready to show the desired result (Theorem 1.1.7):
Corollary 7.0.7. Let 𝑘 be a number field and 𝑆 a finite set of places of 𝑘 containing the
archimedean places. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑥1 , . . . , 𝑥 𝑛 ] be polynomials that don’t both vanish at the
origin (0, . . . , 0). For all 𝜖 > 0, there exist a 𝛿 > 0 and a proper Zariski closed subset 𝑍 ⊂ G𝑛𝑚
such that:
                        ∑︁
                    −         log− max{| 𝑓 (𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 , |𝑔(𝑢 1 , . . . , 𝑢 𝑛 )| 𝑣 } < 𝜖 max ℎ(𝑢𝑖 )
                                                                                                      𝑖
                       𝑣∈𝑀𝑘
for all (𝑢 1 , . . . , 𝑢 𝑛 ) ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 \ 𝑍.
                                                                  44


                                             2
                                   𝜖
Proof. By letting 𝛿 =                              , we obtain the inequality from Theorem 7.0.6. □
                          6𝑛3 (deg 𝑓 + deg 𝑔)
    As discussed in the following remark, under a normal crossings assumption, a result of
Silverman shows that Vojta’s conjecture predicts an improvement to Theorem 7.0.6.
Remark 7.0.8. From Theorem 2 in [18, Silverman], if we assume Vojta’s Conjecture is true,
there is an improvement of the inequality as in Theorem 7.0.3. Let 𝑘 be a number field. Fix
𝜖 > 0. For 𝑓 and 𝑔 homogeneous coprime polynomials in 𝑘 [𝑥0 , . . . , 𝑥 𝑛 ] and 𝑌 = { 𝑓 = 𝑔 = 0}
that intersects the coordinate hyperplanes transversally, there is a proper closed subset 𝑍 such
that we have for all x ∈ P𝑛 (𝑘) \ 𝑍,
                                                                           1    ∑︁
              log gcd( 𝑓 (x), 𝑔(x)) ≤ 𝜖 max{ℎ(𝑥 0 ), . . . , ℎ(𝑥 𝑛 )} +             ℎ ¯ (𝑥𝑖 )
                                                                        1 + 𝛾𝜖 1≤𝑖≤𝑛 𝑆
where 𝛾 is a positive constant.
    Suppose ℎ 𝑆¯ (x) ≤ 𝛿ℎ(x). Using the estimate
                                        ∑︁
                                             ℎ 𝑆¯ (𝑥𝑖 ) ≤ 𝑛ℎ 𝑆¯ (x),
                                       1≤𝑖≤𝑛
we get
                                                 ∑︁                           ∑︁
                                          𝑛𝛿                                𝛿
          log gcd( 𝑓 (x), 𝑔(x)) ≤ 𝜖 +                     ℎ(𝑥𝑖 ) ≤ 𝜖 +           𝑛       ℎ(𝑥𝑖 ).
                                        1 + 𝛾𝜖      1≤𝑖≤𝑛
                                                                         1 + 𝛾𝜖    1≤𝑖≤𝑛
                                                     45


               √︁
          −1 + 1 + 4𝛾𝛿
Let 𝜖 =                   , we obtain a similar inequality as in Theorem 7.0.3,
                2𝛾
                                     ©      √︁                                  ª ∑︁
                                     ­ −1 + 1 + 4𝛾𝛿                  𝛿          ®
           log gcd( 𝑓 (x), 𝑔(x)) ≤ ­ ­                 +             √︁         ®𝑛      ℎ(𝑥𝑖 )
                                     ­       2𝛾               −1 + 1 + 4𝛾𝛿 ® 1≤𝑖≤𝑛
                                                                                ®
                                                           1+
                                     «                               2 ∑︁ ¬
                                           √︁              1   2𝛿
                                  = (−1 + 1 + 4𝛾𝛿)           +         𝑛      ℎ(𝑥𝑖 )
                                                          2𝛾 4𝛾𝛿 1≤𝑖≤𝑛
                                                    
                                                4𝛾𝛿 1 ∑︁
                                  ≤ −1 + 1 +              𝑛     ℎ(𝑥𝑖 )
                                                 2 𝛾 1≤𝑖≤𝑛
                                          ∑︁
                                  = 2𝛿𝑛        ℎ(𝑥𝑖 ).
                                         1≤𝑖≤𝑛
Thus, under a normal crossings assumption, Vojta’s conjecture predicts a linear dependence on
𝛿 in place of the square root dependence in Theorem 7.0.6 (note, however, that without a normal
crossings assumption, the dependence on the degree of 𝑓 and 𝑔 in Theorem 7.0.6 is necessary,
as can be seen by taking high powers of appropriate polynomials).
Example 7.0.9. In this example we show that the predicted linear dependence on 𝛿 is sharp
(if true). Let Q be the field of rationals and 𝑆 = {𝑝, ∞} be a finite set of places in 𝑀Q . Let
0 < 𝛿 < 1. Let 𝑥 = 𝑝 𝑚 , 𝑢 = 𝑝 𝑛 for positive integers 𝑚 and 𝑛 such that 𝑃 := (𝑥, 𝑢(𝑥 + 1)) satisfies
1/2𝛿ℎ(𝑃) ≤ ℎ 𝑆¯ (𝑃) ≤ 𝛿ℎ(𝑃). Let 𝑥 1 , 𝑥2 be the coordinates in G2𝑚 , then we take 𝑓 = 𝑥 1 + 1,
𝑔 = 𝑥 2 . We make the estimate
                             ∑︁                                        ∑︁
                                    −
log gcd( 𝑓 (𝑃), 𝑔(𝑃)) = −        log max{|𝑥 + 1| 𝑣 , |𝑢(𝑥 + 1)| 𝑣 } −      log− max{|𝑥 + 1| 𝑣 , |𝑢(𝑥 + 1)| 𝑣 }
                             𝑣∉𝑆                                       𝑣∈𝑆
                                        
                           ∑︁         1
                        ≥      𝜆𝑣          = ℎ(𝑥 + 1).
                           𝑣∉𝑆
                                    𝑥+1
One shall also notice that for 𝑥 ∈ O𝑆,Q ∗ , we have ℎ (𝑃) = ℎ (𝑥 + 1) = ℎ(𝑥 + 1). Then it follows
                                                       𝑆¯         𝑆¯
that
                                  log gcd( 𝑓 (𝑃), 𝑔(𝑃)) ≥ 1/2𝛿ℎ(𝑃).
                                                    46


It’s easily seen that one may choose infinitely many appropriate 𝑥 and 𝑢 such that the set of
resulting points 𝑃 forms a Zariski dense set in G2𝑚 . Therefore the dependence on 𝛿 has to be at
least linear.
                                               47


                                                   CHAPTER 8
               PROOFS OF LINEAR RECURRENCE SEQUENCES THEOREMS
In this section our main goal is to give the proof of Theorem 1.2.4, which requires Corollary
7.0.4 from the previous section.
Lemma 8.0.1. Let
                                                        ∑︁ 𝑠
                                                𝐹 (𝑛) =      𝑝𝑖 (𝑛)𝛼𝑖𝑛
                                                         𝑖=0
define a nondegenerate algebraic linear recurrence sequence. Let | · | be an absolute value on
Q̄ such that |𝛼𝑖 | ≥ 1 for some 𝑖. Let 0 < 𝜖 < 1. Then
                                                 − log |𝐹 (𝑛)| < 𝜖𝑛
for all but finitely many 𝑛 ∈ N.
Proof. Let 𝑘 be a number field and 𝑆 a finite set of places of 𝑘 such that 𝑝𝑖 (𝑥) ∈ 𝑘 [𝑥], 𝛼𝑖 ∈ O𝑘,𝑆           ∗ ,
𝑖 = 0, . . . , 𝑠, and | · | restricted to 𝑘 is equivalent to | · | 𝑣 for some 𝑣 ∈ 𝑆 (note that if | · | is trivial,
the lemma is obvious). The 𝑠 = 0 case is trivial and so we may assume that 𝑠 > 0. By taking
sufficiently large 𝑛, we can always assume that 𝑝𝑖 (𝑛) don’t vanish simultaneously. It suffices to
prove that
                                                − log |𝐹 (𝑛)| 𝑣 < 𝜖𝑛
for all but finitely many 𝑛 ∈ N.
    Let 𝐻𝑖 be the coordinate hyperplane in P𝑠 defined by 𝑥𝑖 = 0, 𝑖 = 0, . . . , 𝑠. Let 𝐻𝑠+1 be the
hyperplane in P𝑠 defined by 𝑥 0 + 𝑥 1 + · · · + 𝑥 𝑠 = 0. Note that the 𝑠 + 2 hyperplanes 𝐻0 , . . . , 𝐻𝑠+1
                                                          48


are in general position. Let
                                𝑃 = [𝛼0 : · · · : 𝛼𝑠 ] ∈ P𝑠 (𝑘)
                               𝑃𝑛 = [ 𝑝 0 (𝑛)𝛼0𝑛 : · · · : 𝑝 𝑠 (𝑛)𝛼𝑠𝑛 ] ∈ P𝑠 (𝑘), 𝑛 ∈ N
                              𝑄 𝑛 = [ 𝑝 0 (𝑛) : · · · : 𝑝 𝑠 (𝑛)] ∈ P𝑠 (𝑘), 𝑛 ∈ N.
Let ℎ = max{1, ℎ(𝑃)}. Then the Schmidt Subspace Theorem gives that for some finite union of
hyperplanes 𝑍 in P𝑠 ,
                                     𝑠+1
                                     ∑︁
                                         𝑚 𝐻𝑖 ,𝑆 (𝑃𝑛 ) < (𝑠 + 1 + 𝜖/(4ℎ))ℎ(𝑃𝑛 )                            (∗)
                                     𝑖=0
for all points 𝑃𝑛 ∈       P𝑠 (𝑘)   \ 𝑍. In fact, since 𝐹 is nondegenerate, by the Skolem-Mahler-Lech
theorem, only finitely many points 𝑃𝑛 belong to the given hyperplanes in P𝑠 , and thus the
inequality holds for all but finitely many 𝑛. By taking 𝑛 to be sufficiently large, we can assume
                                                𝜖                                                           ∗
that ℎ(𝑄 𝑛 ) ≤ 𝛿ℎ(𝑃𝑛 ) with 𝛿 ≤                         , so that we assume 𝑃𝑛 ∈ G𝑛𝑚 (𝑘) 𝑆,𝛿 . Since 𝛼𝑖 ∈ O𝑘,𝑆
                                           4(𝑠 + 1)ℎ
for all 𝑖, 𝑚 𝐻𝑖 ,𝑆 (𝑃𝑛 ) ≥ (1 − 𝛿)ℎ(𝑃𝑛 ), 𝑖 = 0, . . . , 𝑠. Note also that
                                                                             𝑛
                                      ℎ(𝑃𝑛 ) ≤ 𝑛ℎ(𝑃) + ℎ(𝑄 𝑛 ) ≤                ℎ(𝑃)
                                                                        1−𝛿
for all 𝑛 sufficiently large. Substituting in (∗), we have
                                                                       𝑛                    𝑛𝜖
                        𝑚 𝐻𝑠+1 ,𝑆 (𝑃𝑛 ) < (𝜖/(4ℎ) + (𝑠 + 1)𝛿)                 ℎ(𝑃) ≤                .
                                                                    1−𝛿                2(1 − 𝛿)
                        𝜖                1
Note that 𝛿 =                   ≤               ≤ 1/2, so we have 1 − 𝛿 > 1/2 and then
                   4(𝑠 + 1)ℎ 4(𝑠 + 1)
                                                   𝑚 𝐻𝑠+1 (𝑃𝑛 ) < 𝜖𝑛.
Pick 𝛼 𝑗 with |𝛼 𝑗 | 𝑣 ≥ 1. Then
                          max log | 𝑝𝑖 (𝑛)𝛼𝑖𝑛 | 𝑣 ≥ log | 𝑝 𝑗 (𝑛)| 𝑣 |𝛼 𝑗 | 𝑛𝑣 ≥ log | 𝑝 𝑗 (𝑛)| 𝑣 .
                            𝑖
To give 𝑝 𝑗 (𝑛) an estimate, we can take the inequality
                                             log | 𝑝 𝑗 (𝑛)| 𝑣 ≥ −ℎ( 𝑝 𝑗 (𝑛)).
                                                             49


Then it follows that
                                      log | 𝑝 𝑗 (𝑛)| 𝑣 ≥ − deg 𝑝 𝑗 log 𝑛 + 𝑂 (1).
It follows that
                                              max𝑖 | 𝑝𝑖 (𝑛)𝛼𝑖𝑛 | 𝑣
                      𝜆 𝐻𝑠+1 ,𝑣 (𝑃𝑛 ) = log Í𝑠                      ≥ − log |𝐹 (𝑛)| 𝑣 − 𝐶 ′ log 𝑛
                                              | 𝑖=0 𝑝𝑖 (𝑛)𝛼𝑖𝑛 | 𝑣
for some constant 𝐶 ′. Together with 𝑚 𝐻𝑠+1 ,𝑆 (𝑃𝑛 ) ≥ 𝜆 𝐻𝑠+1 ,𝑣 (𝑃𝑛 ) + 𝑂 (1), we have for all 𝜖 > 0,
                                      − log |𝐹 (𝑛)| 𝑣 < 𝜖𝑛 + 𝐶 ′ log 𝑛 + 𝑂 (1).
It follows that for all sufficiently large 𝑛,
                                                  − log |𝐹 (𝑛)| 𝑣 < 𝜖𝑛.
                                                                                                       □
     Now we can state Theorem 1.8 (i) of Grieve-Wang [11] on the greatest common divisor
between the terms of two linear recurrence sequences with the same index and give an alternative
proof:
Theorem 8.0.2. Let
                                                          ∑︁ 𝑠
                                                 𝐹 (𝑚) =        𝑝𝑖 (𝑚)𝛼𝑖𝑚
                                                           𝑖=1
                                                          ∑︁ 𝑡
                                                  𝐺 (𝑛) =       𝑞 𝑗 (𝑛) 𝛽𝑛𝑗
                                                            𝑗=1
define two algebraic linear recurrence sequences, where 𝑝𝑖 and 𝑞 𝑗 are polynomials. Let 𝑘 be
a number field such that all coefficients of 𝑝𝑖 and 𝑞 𝑗 and 𝛼𝑖 , 𝛽 𝑗 are in 𝑘, for 𝑖 = 1, . . . , 𝑠,
𝑗 = 1, . . . , 𝑡. Let
                      𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Let 𝜖 > 0. Then all but finitely many solutions 𝑙 ∈ N of the inequality
                                      ∑︁
                                             − log− max{|𝐹 (𝑙)| 𝑣 , |𝐺 (𝑙)| 𝑣 } > 𝜖𝑙
                                   𝑣∈𝑀𝑘 \𝑆0
                                                            50


lie in one of finitely many nontrivial arithmetic subprogressions:
                                                  𝑎𝑖 𝑡 + 𝑏𝑖 , 𝑡 ∈ N, 𝑖 = 1, . . . , 𝑟
where 𝑎𝑖 , 𝑏𝑖 ∈ N, 𝑎𝑖 ≠ 0, and the linear recurrences 𝐹 (𝑎𝑖 •+𝑏𝑖 ) and 𝐺 (𝑎𝑖 •+𝑏𝑖 ) have a nontrivial
common factor for 𝑖 = 1, . . . , 𝑟. Furthermore, if 𝐹 and 𝐺 are coprime and their roots generate
a torsion-free group, then there are only finitely many solutions to the inequality above.
Proof. We begin with a couple of convenient reductions. First, by considering finitely many
arithmetic progressions in 𝑙, we may reduce to the case where the combined roots of 𝐹 and 𝐺
generate a torsion-free group Γ of rank 𝑟 (in particular, both 𝐹 and 𝐺 are nondegenerate). Let
𝑆 ⊃ 𝑆0 be a finite set of places of 𝑘, containing the archimedean places, such that all coefficients
of 𝑝𝑖 and 𝑞 𝑗 and 𝛼𝑖 , 𝛽 𝑗 are in O𝑘,𝑆        ∗ for all 𝑖 and 𝑗.
By Lemma 8.0.1,
                                          ∑︁                                                          𝜖
                                                  − log− max{|𝐹 (𝑙)| 𝑣 , |𝐺 (𝑙)| 𝑣 } ≤                  𝑙
                                                                                                     2
                                         𝑣∈𝑆\𝑆0
for all but finitely many 𝑙 ∈ N. Thus it suffices to prove the statement of the theorem with the
inequality:
                                          ∑︁
                                                  − log− max{|𝐹 (𝑙)| 𝑣 , |𝐺 (𝑙)| 𝑣 } < 𝜖𝑙.
                                        𝑣∈𝑀𝑘 \𝑆
     Let 𝑢 1 , . . . , 𝑢𝑟 be generators for Γ. Let 𝑓 , 𝑔 ∈ 𝑘 [𝑙, 𝑥1 , . . . , 𝑥𝑟 , 𝑥1−1 , . . . , 𝑥𝑟−1 ] be the Laurent
polynomials corresponding to 𝐹 and 𝐺. We may write
                                       𝑓 (𝑙, 𝑥1 , . . . , 𝑥𝑟 ) = 𝑥 1𝑖1 · · · 𝑥𝑟𝑖𝑟 𝑓0 (𝑙, 𝑥1 , . . . , 𝑥𝑟 ),
                                                                    𝑗           𝑗
                                      𝑔(𝑙, 𝑥1 , . . . , 𝑥𝑟 ) = 𝑥 11 · · · 𝑥𝑟 𝑟 𝑔0 (𝑙, 𝑥1 , . . . , 𝑥𝑟 )
where 𝑖1 , . . . , 𝑖𝑟 , 𝑗1 , . . . , 𝑗𝑟 ∈ Z and 𝑓0 ∈ 𝑘 [𝑙, 𝑥1 , . . . , 𝑥𝑟 ], 𝑔0 ∈ 𝑘 [𝑙, 𝑥1 , . . . , 𝑥𝑟 ] with 𝑥𝑖 ∤ 𝑓0 𝑔0 ,
𝑖 = 1, . . . , 𝑟. Let 𝐹0 and 𝐺 0 be the linear recurrence sequences corresponding to 𝑓0 and 𝑔0 ,
respectively. Since 𝑢 1 , . . . , 𝑢𝑟 ∈ O𝑘,𝑆         ∗ , it follows that
              ∑︁                                                           ∑︁
                       − log− max{|𝐹 (𝑙)| 𝑣 , |𝐺 (𝑙)| 𝑣 } =                         − log− max{|𝐹0 (𝑙)| 𝑣 , |𝐺 0 (𝑙)| 𝑣 }.
           𝑣∈𝑀𝑘 \𝑆                                                      𝑣∈𝑀𝑘 \𝑆
                                                                      51


Then it suffices to prove the statement of the theorem with 𝐹 and 𝐺 replaced by 𝐹0 and 𝐺 0 ,
respectively. Note that since 𝑥 1 , . . . , 𝑥𝑟 are units in 𝑘 [𝑥 1 , . . . , 𝑥𝑟 , 𝑥1−1 , . . . , 𝑥𝑟−1 ], replacing 𝐹 and
𝐺 by 𝐹0 and 𝐺 0 has no effect on coprimality statements. Thus, we now assume that 𝐹 and 𝐺
correspond to polynomials 𝑓 and 𝑔 in 𝑘 [𝑙, 𝑥1 , . . . , 𝑥𝑟 ] .
     Suppose now that 𝐹 and 𝐺 are coprime (equivalently, 𝑓 and 𝑔 are coprime). Let
                                              𝑃𝑛 = (𝑛, 𝑢 1𝑛 , . . . , 𝑢𝑟𝑛 ).
Now for a fixed sufficiently small positive 𝛿 (coming from the proof of Corollary 7.0.4), take 𝑛
to be sufficiently large such that ℎ(𝑛) ≤ 𝛿𝑛 min ℎ(𝑢𝑖 ), and so 𝑃𝑛 ∈ G𝑟+1               𝑚 (𝑘) 𝑆,𝛿 .
                                                         𝑖
By Corollary 7.0.4,
                       ∑︁
                             − log− max{| 𝑓 (𝑃𝑛 )| 𝑣 , |𝑔(𝑃𝑛 )| 𝑣 } < 𝜖 max{ℎ(𝑢 1𝑛 ), . . . , ℎ(𝑢𝑟𝑛 )}
                     𝑣∈𝑀𝑘 \𝑆
for all 𝑃𝑛 ∈ G𝑟+1     𝑚 (𝑘) 𝑆,𝛿 outside a proper Zariski closed set 𝑍. Noting that 𝑓 (𝑃𝑛 ) = 𝐹 (𝑛) and
𝑔(𝑃𝑛 ) = 𝐺 (𝑛), and also that max{ℎ(𝑢 1𝑛 ), . . . , ℎ(𝑢𝑟𝑛 )} = 𝑛 max{ℎ(𝑢 1 ), . . . , ℎ(𝑢𝑟 )}, after possibly
shrinking 𝜖,we can write the above inequality as
                                    ∑︁
                                          − log− max{|𝐹 (𝑛)| 𝑣 , |𝐺 (𝑛)| 𝑣 } < 𝜖𝑛.
                                  𝑣∈𝑀𝑘 \𝑆
Cover the exceptional set 𝑍 by a hypersurface defined by a polynomial 𝐸𝑥𝑐(𝑥 1 , . . . , 𝑥𝑟+1 ) in
𝑘 [𝑥 1 , . . . , 𝑥𝑟+1 ] such that if 𝑃𝑛 ∈ 𝑍 then 𝐸𝑥𝑐(𝑃𝑛 ) = 0. We can view 𝐸𝑥𝑐(𝑃𝑛 ) as terms of a
linear recurrence sequence 𝐸 (𝑛) with 𝐸 non-degenerate. By the Skolem-Mahler-Lech theorem,
there are only finitely many zeros for 𝐸, which completes the proof.
                                                                                                                       □
     Here we deal with a special case when 𝑚 and 𝑛 are algebraically related:
Lemma 8.0.3. Let
                                                            𝑠
                                                           ∑︁
                                              𝐹 (𝑚) =          𝑝𝑖 (𝑚)𝛼𝑖𝑚
                                                           𝑖=1
                                                            52


                                                        ∑︁ 𝑡
                                             𝐺 (𝑛) =         𝑞 𝑗 (𝑛) 𝛽𝑛𝑗
                                                         𝑗=1
be two linear recurrence sequences over a number field 𝑘 and 𝑆 be a finite set of places in 𝑀𝑘
containing archimedean places and 𝑆0 , where 𝑆0 is defined as
                   𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Let 𝐶 ⊂ A2 be an affine irreducible plane curve over 𝑘. If there are infinitely many (𝑚, 𝑛) ∈ 𝐶 (Z)
satisfying the inequality
                         ∑︁
                                − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } > 𝜖 max{𝑚, 𝑛}
                       𝑣∈𝑀𝑘 \𝑆
then 𝐶 is a line over 𝑘. In particular, if 𝑚(𝑡), 𝑛(𝑡) ∈ Z[𝑡] are polynomials that are not linearly
related, then the inequality
                   ∑︁
                         − log− max{|𝐹 (𝑚(𝑡))| 𝑣 , |𝐺 (𝑛(𝑡))| 𝑣 } > 𝜖 max{𝑚(𝑡), 𝑛(𝑡)}
                 𝑣∈𝑀𝑘 \𝑆
has only finitely many solutions 𝑡 ∈ Z.
Remark 8.0.4. Note that if 𝐶 is a line, the solutions are easily classified using Theorem 8.0.2
    The following lemma is a basic fact from linear algebra, we state it without a proof.
Lemma 8.0.5. Let {𝑣 1 , . . . , 𝑣 𝑛 } be a linearly independent subset of a normed vector space 𝑋.
Then there exists a constant 𝑐 > 0 such that for every set of scalars {𝛼1 , . . . , 𝛼𝑛 }:
                              |𝛼1 𝑣 1 + · · · + 𝛼𝑛 𝑣 𝑛 | ≥ 𝑐(|𝛼1 | + · · · + |𝛼𝑛 |).
                ∗                                             ∗                                     ∗ ∗
    Let 𝑇 𝑜𝑟 (Q ) denote the torsion subgroup of Q . Since the height ℎ gives Q /𝑇 𝑜𝑟 (Q ) the
structure of a normed vector space over Q as in Allcock and Vaaler [1], we immediately find:
                                                                                                 ∗
Lemma 8.0.6. Let 𝑢 1 , . . . , 𝑢 𝑛 be multiplicatively independent elements of Q . Then there exists
a constant 𝑐 > 0 such that for all 𝑖1 , . . . , 𝑖 𝑛 ∈ Z,
                                        ℎ(𝑢𝑖11 · · · 𝑢𝑖𝑛𝑛 ) ≥ 𝑐 max |𝑖 𝑗 |.
                                                                   𝑗
                                                          53


    We now prove Lemma 8.0.3.
Proof. Using the same reduction as in the proof of Theorem 8.0.2, we can assume that the roots
of 𝐹 and 𝐺 are 𝑆-units, and by considering finitely many congruence classes, we can assume
that the roots of 𝐹 and 𝐺 generate a torsion free group. Let 𝐶 be the affine curve defined by the
algebraic relation 𝑅(𝑥 1 , 𝑥2 ) = 0, with 𝑅(𝑥 1 , 𝑥2 ) ∈ 𝑘 [𝑥 1 , 𝑥2 ] irreducible. If 𝐶 is not geometrically
irreducible then 𝐶 (𝑘) (and hence 𝐶 (Z)) is finite, and so we further assume 𝐶 is geometrically
irreducible. By Siegel’s Theorem, 𝐶 (Z) is finite unless 𝐶 has genus 0 and 𝐶 has two or fewer
distinct points at infinity, which we now assume. After replacing 𝑘 by a suitable finite extension,
we can parametrize 𝐶 by Laurent polynomials 𝑚(𝑡), 𝑛(𝑡) ∈ 𝑘 [𝑡, 1/𝑡]. Assume that 𝐶 is not a
line, or equivalently, that 𝑚(𝑡) and 𝑛(𝑡) do not satisfy a linear relation.
    Let Γ be the torsion free group generated by the roots of 𝐹 and 𝐺 and let {𝑢 1 , . . . , 𝑢𝑟 } be
generators of Γ. Consider the points
                                         𝑃𝑡 = (𝑡, 𝑢 1𝑚(𝑡) , . . . , 𝑢𝑟𝑚(𝑡) , 𝑢 1𝑛(𝑡) , . . . , 𝑢𝑟𝑛(𝑡) ),
for 𝑡 ∈ 𝑘 where, as we implicitly assume from now on, we have 𝑚(𝑡), 𝑛(𝑡) ∈ Z. Then for some
Laurent polynomials
               𝑓 (𝑥 1 , . . . , 𝑥𝑟+1 ), 𝑔(𝑥1 , 𝑥𝑟+2 , . . . , 𝑥 2𝑟+1 ) ∈ 𝑘 [𝑥 1 , . . . , 𝑥 2𝑟+1 , 𝑥1−1 , . . . , 𝑥 2𝑟+1−1
                                                                                                                           ],
we have 𝐹 (𝑚(𝑡)) = 𝑓 (𝑃𝑡 ) and 𝐺 (𝑛(𝑡)) = 𝑔(𝑃𝑡 ). From the form of 𝑓 and 𝑔, we may write
                                 𝑓 (𝑥 1 , . . . , 𝑥𝑟+1 ) = 𝑥 2𝑖1 · · · 𝑥𝑟+1
                                                                          𝑖𝑟
                                                                                 𝑐(𝑥 1 ) 𝑓¯(𝑥 1 , . . . , 𝑥𝑟+1 ),
                                                              𝑗 1            𝑗
                      𝑔(𝑥 1 , 𝑥𝑟+2 , . . . , 𝑥 2𝑟+1 ) = 𝑥𝑟+2        · · · 𝑥 2𝑟+1
                                                                               𝑟
                                                                                   𝑐(𝑥 1 ) 𝑔(𝑥
                                                                                            ¯ 1 , 𝑥𝑟+2 , . . . , 𝑥 2𝑟+1 )
where 𝑖1 , . . . , 𝑖𝑟 , 𝑗1 , . . . , 𝑗𝑟 ∈ Z, 𝑓¯ and 𝑔¯ are coprime polynomials in 𝑘 [𝑥 1 , . . . , 𝑥 2𝑟+1 ], and 𝑐(𝑥 1 )
is a Laurent polynomial in 𝑥1 .
    By elementary properties of heights, if 𝑚(𝑡), 𝑛(𝑡) ∈ Z, then ℎ(𝑡) ≪ log max{|𝑚(𝑡)|, |𝑛(𝑡)|}
and ℎ(𝑃𝑡 ) ≫ max{|𝑚(𝑡)|, |𝑛(𝑡)|}. It follows that for any 𝛿 > 0, we have 𝑃𝑡 ∈ G2𝑟+1                                      𝑚 (𝑘) 𝑆,𝛿 for all
                                                                       54


but finitely many 𝑡 ∈ 𝑘 (with 𝑚(𝑡), 𝑛(𝑡) ∈ Z). Then Corollary 7.0.4 applies to 𝑓¯ and 𝑔¯ and we
obtain that for any 𝜖 > 0 there exists a proper Zariski closed subset 𝑍 ⊂ G2𝑟+1                           𝑚    such that
                  ∑︁
                        − log− max{| 𝑓¯(𝑃𝑡 )| 𝑣 , | 𝑔(𝑃  ¯ 𝑡 )| 𝑣 } < 𝜖 max {ℎ(𝑢𝑖𝑚(𝑡) ), ℎ(𝑢𝑖𝑛(𝑡) )}
                                                                             𝑖=1,...,𝑟
                𝑣∈𝑀𝑘 \𝑆
for all points 𝑃𝑡 outside 𝑍. By elementary estimates, for all but finitely many 𝑡 ∈ 𝑘,
                      ∑︁
                             − log− |𝑐(𝑡)| 𝑣 ≤ ℎ(𝑐(𝑡)) < 𝜖 max {ℎ(𝑢𝑖𝑚(𝑡) ), ℎ(𝑢𝑖𝑛(𝑡) )}.
                                                                       𝑖=1,...,𝑟
                    𝑣∈𝑀𝑘 \𝑆
     Using this inequality and that 𝑢 1 , . . . , 𝑢𝑟 ∈ O𝑘,𝑆           ∗ , the inequality for 𝑓¯ and 𝑔¯ implies the
inequality for 𝑓 and 𝑔:
                  ∑︁
                        − log− max{| 𝑓 (𝑃𝑡 )| 𝑣 , |𝑔(𝑃𝑡 )| 𝑣 } < 𝜖 max {ℎ(𝑢𝑖𝑚(𝑡) ), ℎ(𝑢𝑖𝑛(𝑡) )}
                                                                             𝑖=1,...,𝑟
                𝑣∈𝑀𝑘 \𝑆
for all points 𝑃𝑡 outside a proper Zariski closed subset 𝑍 ⊂ G2𝑟+1                𝑚 . Setting (𝑚, 𝑛) = (𝑚(𝑡), 𝑛(𝑡)) ∈
𝐶 (Z), note that 𝑓 (𝑃𝑡 ) = 𝐹 (𝑚), 𝑔(𝑃𝑡 ) = 𝐺 (𝑛), and
        max{ℎ(𝑢 1𝑚 ), . . . , ℎ(𝑢𝑟𝑚 ), ℎ(𝑢 1𝑛 ), . . . , ℎ(𝑢𝑟𝑛 )} ≤ max{𝑚, 𝑛} max{ℎ(𝑢 1 ), . . . , ℎ(𝑢𝑟 )}.
Then we can write the above inequality as
                           ∑︁
                                  − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } < 𝜖 max{𝑚, 𝑛}.
                         𝑣∈𝑀𝑘 \𝑆
It remains to show that there are only finitely many 𝑡 ∈ 𝑘 with 𝑚(𝑡), 𝑛(𝑡) ∈ Z and 𝑃𝑡 ∈ 𝑍. Now
we cover 𝑍 by a hypersurface defined by an equation 𝑧(𝑥1 , . . . , 𝑥 2𝑟+1 ) = 0. Then every 𝑃𝑡 in 𝑍
satisfies an equation
                                𝐾
                               ∑︁             𝑚(𝑡)𝑠1,𝑤           𝑚(𝑡)𝑠𝑟 ,𝑤 𝑛(𝑡)𝑡1,𝑤             𝑛(𝑡)𝑡𝑟 ,𝑤
                    𝑧(𝑃𝑡 ) =       𝑃𝑤 (𝑡)𝑢 1             · · · 𝑢𝑟         𝑢1           · · · 𝑢𝑟           = 0,
                               𝑤=1
where 𝑃𝑤 ∈ 𝑘 [𝑡], 𝑤 = 1, . . . , 𝐾 are nonzero polynomials and the integer tuples (𝑠1,𝑤 , . . . , 𝑠𝑟,𝑤 , 𝑡1,𝑤 , . . . , 𝑡𝑟,𝑤 ),
𝑤 = 1, . . . , 𝐾, are distinct. If 𝐾 = 1 then 𝑡 must be one of the finitely many roots of the polyno-
mial 𝑃1 (𝑡). Otherwise, dividing by the first term we find
                              𝐾                   ′ +𝑛(𝑡)𝑡 ′
                                                                         𝑚(𝑡)𝑠𝑟′ ,𝑤 +𝑛(𝑡)𝑡𝑟′ ,𝑤
                            ∑︁            𝑚(𝑡)𝑠1,𝑤           1,𝑤
                                 𝑄 𝑤 (𝑡)𝑢 1                      · · · 𝑢𝑟                        = 1,                    (8.1)
                            𝑤=2
                                                              55


                                                                                ′ = 𝑠                      ′
where 𝑄 𝑤 (𝑡), 𝑖 = 2, . . . , 𝐾, are rational functions in 𝑡 and 𝑠𝑖,𝑤                       𝑖,𝑤 − 𝑠𝑖,1 , 𝑡𝑖,𝑤 = 𝑡𝑖,𝑤 − 𝑡𝑖,1 .
    Note that
                                     ℎ(𝑄 𝑤 (𝑡)) = (deg 𝑄 𝑤 )ℎ(𝑡) + 𝑂 (1)
and by Lemma 8.0.6 (assuming 𝑚(𝑡), 𝑛(𝑡) ∈ Z as usual)
                   𝑚(𝑡)𝑠 ′ +𝑛(𝑡)𝑡 ′          𝑚(𝑡)𝑠 ′ +𝑛(𝑡)𝑡𝑟′ ,𝑤
                                                                   
                                                                                               ′             ′
                ℎ 𝑢 1 1,𝑤          1,𝑤
                                       · · · 𝑢𝑟 𝑟 ,𝑤                  ≫ max{|𝑚(𝑡)𝑠𝑖,𝑤              + 𝑛(𝑡)𝑡𝑖,𝑤  |}
                                                                              𝑖
                                                                                             ′       ′
                                                                      = 𝑒 max𝑖 ℎ(𝑚(𝑡)𝑠𝑖,𝑤 +𝑛(𝑡)𝑡𝑖,𝑤 )
                                                                                                 ′       ′
                                                                      ≫ 𝑒 ℎ(𝑡) max𝑖 deg(𝑚𝑠𝑖,𝑤 +𝑛𝑡𝑖,𝑤 )
                                                                      ≫ 𝑒 ℎ(𝑡)
         ′ , 𝑡 ′ ) ≠ (0, 0) for some 𝑖, and in this case 𝑚𝑠′ + 𝑛𝑡 ′ must be nonconstant by our
since (𝑠𝑖,𝑤   𝑖,𝑤                                                            𝑖,𝑤          𝑖,𝑤
assumption that 𝑚 and 𝑛 aren’t linearly related.
    Since the terms in the sum in (8.1) are 𝑆-units outside the factors 𝑄 𝑤 (𝑡), it follows from the
height estimates above and the almost 𝑆-unit equation (Corollary 6.2.2) that there exists a finite
set F ⊂ 𝑘 such that every solution 𝑡 ∈ 𝑘 to (8.1) (with 𝑚(𝑡), 𝑛(𝑡) ∈ Z) satisfies
                                              ′ +𝑛(𝑡)𝑡 ′
                                        𝑚(𝑡)𝑠1,𝑤        1,𝑤         𝑚(𝑡)𝑠𝑟′ ,𝑤 +𝑛(𝑡)𝑡𝑟′ ,𝑤
                              𝑄 𝑤 (𝑡)𝑢 1                     · · · 𝑢𝑟                        ∈F
for some 𝑤. By the height estimates above,
                                             ′ +𝑛(𝑡)𝑡 ′
                                      𝑚(𝑡)𝑠1,𝑤        1,𝑤          𝑚(𝑡)𝑠𝑟′ ,𝑤 +𝑛(𝑡)𝑡𝑟′ ,𝑤
                        ℎ(𝑄 𝑤 (𝑡)𝑢 1                       · · · 𝑢𝑟                       ) ≫ 𝑒 ℎ(𝑡) ,
and Northcott’s Theorem implies that there are only finitely many solutions 𝑡 ∈ 𝑘 with
𝑚(𝑡), 𝑛(𝑡) ∈ Z satisfying (8.1). It follows that there are only finitely many pairs (𝑚, 𝑛) ∈ 𝐶 (Z)
satisfying the inequality of the theorem.
                                                                                                                              □
Definition 8.0.7. Let 𝐹 and 𝐺 be two linear recurrence sequences. Suppose that the roots of
𝐹 and 𝐺 generate multiplicative torsion-free groups of rank 𝑟 and 𝑠, respectively. We say that
the roots of 𝐹 and 𝐺 are multiplicatively independent if the combined roots generate a group of
rank 𝑟 + 𝑠. Otherwise, we say they are multiplicatively dependent.
                                                            56


    The following result is a generalization of Theorem 8.0.2 under a multiplicative independence
assumption, which was proved by Grieve-Wang [11]. Here we give an alternative proof:
Theorem 8.0.8. Let
                                                                  ∑︁ 𝑠
                                                     𝐹 (𝑚) =            𝑝𝑖 (𝑚)𝛼𝑖𝑚
                                                                   𝑖=1
                                                                   ∑︁𝑡
                                                      𝐺 (𝑛) =           𝑞 𝑗 (𝑛) 𝛽𝑛𝑗
                                                                    𝑗=1
define two algebraic linear recurrence sequences, where 𝑝𝑖 and 𝑞 𝑗 are polynomials. Let 𝑘 be
a number field such that all coefficients of 𝑝𝑖 and 𝑞 𝑗 and 𝛼𝑖 , 𝛽 𝑗 are in 𝑘, for 𝑖 = 1, . . . , 𝑠,
𝑗 = 1, . . . , 𝑡. Let
                          𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Let 𝜖 > 0. If we assume further the roots of 𝐹 and 𝐺 are independent, then all but finitely many
(𝑚, 𝑛) ∈ N2 satisfy the inequality
                                   ∑︁
                                            − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } < 𝜖 max{𝑚, 𝑛}.
                               𝑣∈𝑀𝑘 \𝑆0
    In particular, if 𝑆0 = ∅, then all but finitely many (𝑚, 𝑛) satisfy the inequality
                                             log gcd(𝐹 (𝑚), 𝐺 (𝑛)) < 𝜖 max{𝑚, 𝑛}
Proof. Notice that
                        ∑︁
                                 − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } ≤ min{ℎ(𝐹 (𝑚)), ℎ(𝐺 (𝑛))}
                     𝑣∈𝑀𝑘 \𝑆
                                                                             ≤ K min{𝑚, 𝑛}
for some constant K. Hence for the inequality in the statement to be true, for a fixed 𝜖 > 0,
                                                 K min{𝑚, 𝑛} ≥ 𝜖 max{𝑚, 𝑛}.
    The combined roots of 𝐹 and 𝐺 generate a torsion-free group Γ of rank 𝑟 +𝑠 whose generators
are {𝑢 1 , . . . , 𝑢𝑟 , 𝑣 1 , . . . , 𝑣 𝑠 } where 𝑢 1 , . . . , 𝑢𝑟 generate the roots 𝛼𝑖 and 𝑣 1 , . . . , 𝑣 𝑠 generate the
                                                                    57


roots 𝛽 𝑗 . By the same reduction step in the previous proof, assume all the coefficients of the
polynomials 𝑝𝑖 and 𝑞 𝑗 and all of the roots of 𝐹 and 𝐺 are 𝑆-units. We can also assume the
Laurent polynomials 𝑓 and 𝑔 corresponding to 𝐹 and 𝐺 with respect to the roots 𝑢 1 , . . . , 𝑢𝑟 and
𝑣 1 , . . . , 𝑣 𝑠 , respectively, are polynomials.
      Let 𝑓ˆ, 𝑔ˆ ∈ 𝑘 [𝑥 1 , . . . , 𝑥𝑟+𝑠+2 ] be polynomials such that
                                            𝑓ˆ(𝑥 1 , . . . , 𝑥𝑟+𝑠+2 ) = 𝑓 (𝑥 1 , . . . , 𝑥𝑟+1 )
                                         ˆ 1 , . . . , 𝑥𝑟+𝑠+2 ) = 𝑔(𝑥𝑟+2 , . . . , 𝑥𝑟+𝑠+2 ).
                                        𝑔(𝑥
Note that 𝑓ˆ and 𝑔ˆ are coprime since they involve disjoint sets of variables.
      For 𝑚, 𝑛 ∈ N, let
                                          𝑃𝑚,𝑛 = (𝑚, 𝑢 1𝑚 , . . . , 𝑢𝑟𝑚 , 𝑛, 𝑣 1𝑛 , . . . , 𝑣 𝑛𝑠 ).
Let 𝜖 > 0 and let 𝛿 > 0 be the quantity from Corollary 7.0.4 for 𝑓ˆ, 𝑔,                              ˆ and 𝜖. After excluding
finitely many pairs (𝑚, 𝑛), we can always assume that
                                   ℎ(𝑚) + ℎ(𝑛) < 𝛿(𝑟 + 𝑠 + 2) max{ℎ(𝑢𝑖𝑚 ), ℎ(𝑣 𝑛𝑗 )}.
                                                                           𝑖, 𝑗
Therefore 𝑃𝑚,𝑛 ∈ G𝑟+𝑠+2      𝑚      (𝑘) 𝑆,𝛿 . Applying Corollary 7.0.4,
       ∑︁
                − log− max{| 𝑓ˆ(𝑃𝑚,𝑛 )| 𝑣 , | 𝑔(𝑃 ˆ 𝑚,𝑛 )| 𝑣 } < 𝜖 max{ℎ(𝑢 1𝑚 ), . . . , ℎ(𝑢𝑟𝑚 ), ℎ(𝑣 1𝑛 ), . . . , ℎ(𝑣 𝑛𝑠 )}
     𝑣∈𝑀𝑘 \𝑆
for all 𝑃𝑚,𝑛 ∈ G𝑟+𝑠+2      𝑚      (𝑘) 𝑆,𝛿 outside a proper Zariski closed set 𝑍 ⊂ G𝑟+𝑠+2                     𝑚      . Noting that
 𝑓ˆ(𝑃𝑚,𝑛 ) = 𝐹 (𝑚), 𝑔(𝑃    ˆ 𝑚,𝑛 ) = 𝐺 (𝑛), and
                          max        {ℎ(𝑢𝑖𝑚 ), ℎ(𝑣 𝑛𝑗 )} ≤ max{𝑛, 𝑚}                 max          {ℎ(𝑢𝑖 ), ℎ(𝑣 𝑗 )},
                       1≤𝑖≤𝑟,1≤ 𝑗 ≤𝑠                                            1≤𝑖≤𝑟,1≤ 𝑗 ≤𝑠
we can write the above inequality as
                                ∑︁
                                        − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } < 𝜖 max{𝑛, 𝑚}.
                              𝑣∈𝑀𝑘 \𝑆
                                                                    58


As in the 𝑚 = 𝑛 case, we cover 𝑍 by a hypersurface defined by a polynomial equation:
                                                   𝐸𝑥𝑐(𝑥 1 , . . . , 𝑥𝑟+𝑠+2 ) = 0.
Hence all the points 𝑃𝑚,𝑛 in 𝑍 must satisfy the above equation. Therefore, if 𝑃𝑚,𝑛 ∈ 𝑍, after
combining the terms with the same exponents on 𝑢 1 , . . . , 𝑢𝑟 , 𝑣 1 , . . . , 𝑣 𝑠 , we obtain an equation
in terms of 𝑚, 𝑛, 𝑢 1 , . . . , 𝑢𝑟 , 𝑣 1 , . . . , 𝑣 𝑠 :
                                                                 𝑊
                                                                ∑︁                      𝑚𝑠1,𝑤            𝑚𝑠𝑟 ,𝑤 𝑛𝑡1,𝑤            𝑛𝑡 𝑠,𝑤
      𝐸𝑥𝑐(𝑚, 𝑢 1𝑚 , . . . , 𝑢𝑟𝑚 , 𝑛, 𝑣 1𝑛 , . . . , 𝑣 𝑛𝑠 )  =         𝑃𝑤 (𝑚, 𝑛)𝑢 1              · · · 𝑢𝑟        𝑣1        · · · 𝑣𝑠      = 0,
                                                                𝑤=1
where 𝑃𝑤 (𝑚, 𝑛) is a non-zero polynomial in 𝑚 and 𝑛. It follows from Theorem 8.0.2 and Lemma
8.0.3 that after excluding finitely many pairs (𝑚, 𝑛) we can assume that (𝑚, 𝑛) is not a zero of
any of the polynomials 𝑃𝑤 .
    Dividing both sides by the negative of the first term,
                                                        𝑚𝑠               𝑚𝑠       𝑛𝑡              𝑛𝑡
                               𝑊
                              ∑︁    𝑃𝑤 (𝑚, 𝑛)𝑢 1 1,𝑤 · · · 𝑢𝑟 𝑟 ,𝑤 𝑣 1 1,𝑤 · · · 𝑣 𝑠 𝑠,𝑤
                                                           𝑚𝑠1,1          𝑚𝑠𝑟 ,1 𝑛𝑡 1,1           𝑛𝑡 𝑠,1   = 1.
                              𝑤=2   −𝑃1 (𝑚, 𝑛)𝑢 1                  · · · 𝑢𝑟      𝑣1       · · · 𝑣𝑠
                     𝑃𝑤 (𝑚, 𝑛)
Let 𝑄 𝑤 (𝑚, 𝑛) =                     (𝑤 = 2, . . . , 𝑊), then
                    −𝑃1 (𝑚, 𝑛)
                𝑊
               ∑︁                   𝑚(𝑠1,𝑤 −𝑠1,1 )               𝑚(𝑠𝑟 ,𝑤 −𝑠𝑟 ,1 ) 𝑛(𝑡1,𝑤 −𝑡1,1 )           𝑛(𝑡 𝑠,𝑤 −𝑡 𝑠,1 )
                    𝑄 𝑤 (𝑚, 𝑛)𝑢 1                        · · · 𝑢𝑟                𝑣1                 · · · 𝑣𝑠                 = 1.
               𝑤=2
          ′ = 𝑠                   ′
Letting 𝑠𝑖,𝑤      𝑖,𝑤 − 𝑠𝑖,1 , 𝑡𝑖,𝑤 = 𝑡𝑖,𝑤 − 𝑡𝑖,1 , we have
                                𝑊                             ′                       ′
                                                                          𝑚𝑠𝑟′ ,𝑤 𝑛𝑡 1,𝑤              ′
                              ∑︁                          𝑚𝑠1,𝑤                                    𝑛𝑡 𝑠,𝑤
                                    𝑄 𝑤 (𝑚, 𝑛)𝑢 1                 · · · 𝑢𝑟       𝑣1        · · · 𝑣𝑠        =1
                              𝑤=2
       ′ , 𝑡 ′ fixed and only depending on 𝐸𝑥𝑐.
with 𝑠𝑖,𝑤   𝑖,𝑤
    As in the proof of Lemma 8.0.3, it follows from Lemma 8.0.6 that if min{𝑚, 𝑛} is sufficiently
large, then Corollary 6.2.2 applies to the equation
                               𝑊                             ′                       ′
                                                                         𝑚𝑠𝑟′ ,𝑤 𝑛𝑡 1,𝑤              ′
                              ∑︁                         𝑚𝑠1,𝑤                                    𝑛𝑡 𝑠,𝑤
                                    𝑄 𝑤 (𝑚, 𝑛)𝑢 1                · · · 𝑢𝑟        𝑣1       · · · 𝑣𝑠        = 1,
                              𝑤=2
                                                                       59


and we conclude that one of the summands on the left-hand side belongs to a finite set F . But
since
                                  ′                     ′
                               𝑚𝑠1,𝑤         𝑚𝑠𝑟′ ,𝑤 𝑛𝑡 1,𝑤            ′
                                                                    𝑛𝑡 𝑠,𝑤
                 ℎ(𝑄 𝑤 (𝑚, 𝑛)𝑢 1      · · · 𝑢𝑟      𝑣1       · · · 𝑣𝑠      ) → ∞ as min{𝑚, 𝑛} → ∞,
and min{𝑚, 𝑛} → ∞ also means 𝑚𝑎𝑥{𝑚, 𝑛} → ∞ by the remarks at the beginning of the proof,
this implies that there are only finitely many possibilities for the pair (𝑚, 𝑛).                           □
    We now prove a result in the general case where the roots of 𝐹 and 𝐺 are not necessarily
independent. The following theorem gives an improvement to Theorem 1.8 (ii) of Grieve-Wang
[11], who proved a similar result but with log max{𝑚, 𝑛} replaced by the weaker expression
𝑜(max{𝑚, 𝑛}).
Theorem 8.0.9. Let
                                                           ∑︁ 𝑠
                                               𝐹 (𝑚) =             𝑝𝑖 (𝑚)𝛼𝑖𝑚
                                                            𝑖=1
                                                            ∑︁ 𝑡
                                               𝐺 (𝑛) =             𝑞 𝑗 (𝑛) 𝛽𝑛𝑗
                                                             𝑗=1
define two distinct algebraic linear recurrence sequences, where 𝑝𝑖 and 𝑞 𝑗 are polynomials. Let
𝑘 be a number field such that all coefficients of 𝑝𝑖 and 𝑞 𝑗 and 𝛼𝑖 , 𝛽 𝑗 are in 𝑘, for 𝑖 = 1, . . . , 𝑠,
𝑗 = 1, . . . , 𝑡. Let
                      𝑆0 = {𝑣 ∈ 𝑀𝑘 : max{|𝛼1 | 𝑣 , . . . , |𝛼𝑠 | 𝑣 , |𝛽1 | 𝑣 , . . . , |𝛽𝑡 | 𝑣 } < 1}.
Then there are finitely many choices of nonzero integers (𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , 𝑑𝑖 ), 𝑎𝑖 𝑐𝑖 ≠ 0 such that all
solutions (𝑚, 𝑛) ∈ N2 of the inequality
                           ∑︁
                                  − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } > 𝜖 max{𝑚, 𝑛}                        (△)
                         𝑣∈𝑀𝑘 \𝑆0
are of the form:
           (𝑚, 𝑛) = (𝑎𝑖 𝑡 + 𝑏𝑖 , 𝑐𝑖 𝑡 + 𝑑𝑖 ) + (𝜇1 , 𝜇2 ), |𝜇1 |, |𝜇2 | ≪ log 𝑡, 𝑡 ∈ N, 𝑖 = 1, . . . , 𝑟.
                                                             60


Proof. Now let {𝑢 1 , . . . , 𝑢𝑟 } be a set of generators which generates the roots of 𝐹 and 𝐺 and
assume that the 𝑢𝑖 ’s are multiplicatively independent (as in the proof of Theorem 8.0.2). It follows
from the first part of the proof of Theorem 8.0.8 (using the points 𝑃𝑚,𝑛 = (𝑚, 𝑢 1𝑚 , ..., 𝑢𝑟𝑚 , 𝑛, 𝑢 1𝑛 , ..., 𝑢𝑟𝑛 ))
that all but finitely many pairs (𝑚, 𝑛) that fail the above inequality either satisfy finitely many
linear relations (𝑚, 𝑛) = (𝑎𝑖 𝑡 + 𝑏𝑖 , 𝑐𝑖 𝑡 + 𝑑𝑖 ) or satisfy an exponential-polynomial equation coming
from Schmidt’s Subspace Theorem:
                                 𝑊
                                ∑︁                     𝑚𝑠1,𝑤 +𝑛𝑡 1,𝑤          𝑚𝑠𝑟 ,𝑤 +𝑛𝑡𝑟 ,𝑤
                                      𝑃𝑤 (𝑚, 𝑛)𝑢 1                    · · · 𝑢𝑟                  = 0,
                                𝑤=1
where 𝑃𝑤 (𝑚, 𝑛) are non-zero polynomials in 𝑚 and 𝑛. After ignoring finitely many arithmetic
progressions, we can assume that (𝑚, 𝑛) is not a zero of any 𝑃𝑤 by Lemma 8.0.3.
    Dividing by the first term, we need to study the solutions (𝑚, 𝑛) to the equation
                                  𝑊                       ′ +𝑛𝑡 ′
                                                                               𝑚𝑠𝑟′ ,𝑤 +𝑛𝑡𝑟′ ,𝑤
                                ∑︁                     𝑚𝑠1,𝑤      1,𝑤
                                      𝑄 𝑤 (𝑚, 𝑛)𝑢 1                    · · · 𝑢𝑟                  =1                      (▲)
                                𝑤=2
where 𝑄 𝑤 = −𝑃𝑤 (𝑚, 𝑛)/𝑃1 (𝑚, 𝑛).
    As in Theorem 8.0.8, we can estimate the non-𝑆 contribution to the height of each term in
(▲) by
                              ℎ(𝑄 𝑤 (𝑚, 𝑛)) ≤ 𝑅𝑤 max{log 𝑚, log 𝑛} + 𝑂 (1)
for some constant 𝑅𝑤 . On the other hand, we have the estimate
                  ′ +𝑛𝑡 ′
                𝑚𝑠1,𝑤    1,𝑤        𝑚𝑠𝑟′ ,𝑤 +𝑛𝑡𝑟′ ,𝑤                          ′          ′
 ℎ(𝑄 𝑤 (𝑚, 𝑛)𝑢 1             · · · 𝑢𝑟                ) ≥ 𝑐 𝑤 max{|𝑚𝑠𝑖,𝑤           + 𝑛𝑡𝑖,𝑤      |} − 𝑅𝑤 log max{𝑚, 𝑛} + 𝑂 (1)
                                                                 𝑖
for some constant 𝑐 𝑤 .
    In order to apply Corollary 6.2.2, we need each summand to be in 𝑘 𝑆,𝛿 for some 𝛿 <
     1
            . So it suffices to require, for every 𝑤,
𝑊 (𝑊 + 1)
                                                                                     ′           ′
                              𝐶𝑤 max{log 𝑚, log 𝑛} ≤ max{|𝑚𝑠𝑖,𝑤                          + 𝑛𝑡𝑖,𝑤   |}                   (⋆)
                                                                        𝑖
                                                              61


              4𝑅𝑤 𝑊 (𝑊 + 1)
where 𝐶𝑤 =                          . For those (𝑚, 𝑛) satisfying (⋆), we can apply Corollary 6.2.2 to
                        𝑐𝑤
(▲). But since
                                      ′ +𝑛𝑡 ′
                                    𝑚𝑠1,𝑤  1,𝑤        𝑚𝑠𝑟′ ,𝑤 +𝑛𝑡𝑟′ ,𝑤
                ℎ(𝑄 𝑤 (𝑚, 𝑛)𝑢 1                · · · 𝑢𝑟                ) → ∞ as max{𝑚, 𝑛} → ∞,
this implies that there are only finitely many solutions (𝑚, 𝑛) of
                              ∑︁
                                    − log− max{|𝐹 (𝑚)| 𝑣 , |𝐺 (𝑛)| 𝑣 } > 𝜖 max{𝑚, 𝑛}
                           𝑣∈𝑀𝑘 \𝑆0
satisfying (⋆).
    For pairs (𝑚, 𝑛) not satisfying (⋆), there exists some 𝑤 0 and 𝑖0 such that (𝑠𝑖′0 ,𝑤 0 , 𝑡𝑖′0 ,𝑤 0 ) ≠
(0, 0) and
                                 𝐶𝑤 0 max{log 𝑚, log 𝑛} ≥ |𝑚𝑠𝑖′0 ,𝑤 0 + 𝑛𝑡𝑖′0 ,𝑤 0 |.
In fact, since as previously observed, min{𝑚, 𝑛} ≫ max{𝑚, 𝑛} for solutions (𝑚, 𝑛) to (△), we
may assume 𝑠𝑖′0 ,𝑤 0 𝑡𝑖′0 ,𝑤 0 ≠ 0.
    Fix such a pair (𝑚, 𝑛) and corresponding 𝑤 0 and 𝑖0 . Let 𝑎 = 𝑠𝑖′0 ,𝑤 0 , 𝑏 = 𝑡𝑖′0 ,𝑤 0 , and 𝑡 =
         
max 𝑚𝑏 , − 𝑎𝑛 . Replacing (𝑎, 𝑏) by (−𝑎, −𝑏) if necessary, we may assume that 𝑎 < 0 and
𝑏 > 0. We set 𝜇1 = 𝑚 − 𝑏𝑡 and 𝜇2 = 𝑛 + 𝑎𝑡, so that (𝑚, 𝑛) = (𝑏𝑡, −𝑎𝑡) + (𝜇1 , 𝜇2 ). Then clearly
min{|𝜇1 |, |𝜇2 |} ≤ max{|𝑎|, |𝑏|} and so
            max{|𝜇1 |, |𝜇2 |} ≪ |𝑎𝜇1 + 𝑏𝜇2 | = |𝑎𝑚 + 𝑏𝑛| ≪ max{log 𝑚, log 𝑛} ≪ log 𝑡
as desired.                                                                                              □
                                                            62


                                            CHAPTER 9
                     QUADRATIC POINTS ON ABELIAN SURFACES
Let 𝐶 be a curve of genus 2 over a number field 𝑘, it is necessarily a hyperelliptic curve and we
denote its hyperelliptic involution by 𝜎. Let 𝑃0 be a rational point on 𝐶, after possibly enlarging
the number field 𝑘, and consider the embedding 𝑗 : 𝐶 → 𝐽 (𝐶), 𝑃 ↦→ [𝑃 − 𝑃0 ], where [𝑃 − 𝑃0 ]
denote the divisor class of 𝑃 − 𝑃0 on 𝐶.
    By Song-Tucker [19],
Theorem 9.0.1. For any 𝜖 > 0, there exists a constant 𝑂 𝜖 (1) such that for all 𝑃 ∈ 𝐶 ( 𝑘)     ¯ of
degree 𝑑 over 𝑘 with ℎ0 (𝐶, 𝑃 [1] + · · · + 𝑃 [𝑑] ) = 1, we have
                           𝑑 𝑎 (𝑃) ≤ ℎ 𝐾 (𝑃) + (2𝑑 − 2 + 𝜖)ℎ(𝑃) + 𝑂 𝜖 (1),
where 𝑃𝑖 are the conjugates of 𝑃.
    Let 𝑃 be a quadratic point over 𝑘 on 𝐶 with 𝜏 the nontrivial element of the Galois group of
𝑘 (𝑃) over 𝑘, suppose that (𝑃, 𝜏𝑃) ≠ (𝑃, 𝜎𝑃). Then 𝑃 + 𝜏𝑃 ≁ 2𝑃0 , hence dim |𝑃 + 𝜏𝑃| = 0.
The above theorem tells us
                     𝑑 𝑎 (𝑃) ≤ ℎ 𝐾 (𝑃) + (2 + 𝜖)ℎ(𝑃) + 𝑂 𝜖 (1) ≤ (4 + 𝜖)ℎ(𝑃).
On the other hand, we have
                                       𝜙 : 𝐶 2 → 𝐶 (2) → 𝐽 (𝐶)
with the first map being the quotient by 𝑆2 and the second map being the blow up along the point
0. In Silverman’s definition of generalized GCD [18], let 𝑋 be a variety and 𝑌 a subvariety of
codimension at least 2, then let 𝑋˜ be the blow up of 𝑋 along Y and 𝑌˜ be the exceptional divisor.
Let 𝑃˜ be the preimage of 𝑃 in 𝑋. ˜ Then the generalized GCD of a point 𝑃 ∈ (𝑋 \ 𝑌 ) with respect
to 𝑌 is
                                        ℎ𝑔𝑐𝑑 (𝑃; 𝑌 ) = ℎ 𝑋,      ˜
                                                          ˜ 𝑌˜ ( 𝑃).
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Followed by this definition, we can conjecture the GCD inequality on abelian surfaces, which is
a consequence of Vojta’s conjecture.
Conjecture 9.0.2. Let 𝑋 be an abelian surface and 𝐴 an ample divisor on 𝑋, then for 𝑃 ∈ 𝑋 we
have
                                       ℎ𝑔𝑐𝑑 (𝑃) ≤ 𝜖 ℎ 𝐴 (𝑃) + 𝑂 (1)
for all points outside a Zariski closed proper subset 𝑍 of 𝑋.
    It is well-known that all abelian varieties are quotients of Jacobian varieties, in particular,
almost all abelian surfaces come from 𝐽 (𝐶) of a curve 𝐶 of genus 2. Then the question turns
to whether the above inequality holds on 𝐽 (𝐶). If we consider the rational points on 𝐽 (𝐶), and
they pull back to rational points on 𝐶 (2) and all but finitely many pull back to quadratic points
on 𝐶 2 . By pulling everything back to 𝐶 2 , the conjecture is equivalent to the following inequality
on 𝐶, for {𝑃 ∈ 𝐶 |[𝑘 (𝑃) : 𝑘] = 2, 𝜎𝑃 ≠ 𝜏𝑃},
                                       ℎΔ 𝛼 (𝑃) ≤ 𝜖 ℎ 𝐴 (𝑃) + 𝑂 (1)
where Δ𝛼 = {(𝑃, 𝜎𝑃)|𝑃 ∈ 𝐶}. Note that we have the relation between arithmetic discriminant
and heights:
                        𝑑 𝑎 (𝑃) = ℎ 𝐾 (𝑃) + 4ℎ(𝑃) − ℎ 𝜙∗ Θ (𝑃 [1] , 𝑃 [2] ) + 𝑂 (1)            (9.1)
where Θ is the theta divisor defined by Θ = 𝑗 (𝐶) with 𝑗 the map 𝑗 : 𝐶 → 𝐽 (𝐶) via 𝑃 ↦→ [𝑃−𝑃0 ].
Since
                                   𝜙∗ Θ = Δ𝛼 + {𝑃0 } × 𝐶 + 𝐶 × {𝑃0 }
then if one assumes Conjecture 9.0.2 is true, then
                              𝑑 𝑎 (𝑃) = 4ℎ(𝑃) − ℎΔ 𝛼 (𝑃 [1] , 𝑃 [2] ) + 𝑂 (1)
                                       ≥ 4ℎ(𝑃) − 𝜖 ℎ(𝑃) + 𝑂 (1)
                                       ≥ (4 − 𝜖)ℎ(𝑃).
Hence in this case one should expect the conjecture
                                                   64


Conjecture 9.0.3. Notations are as before. For all but finitely many quadratic points 𝑃 with
(𝑃, 𝜏𝑃) ≠ (𝑃, 𝜎𝑃), then if the GCD conjecture is true, we have
                                (4 − 𝜖)ℎ(𝑃) ≤ 𝑑 𝑎 (𝑃) ≤ (4 + 𝜖)ℎ(𝑃).
    Conjecture 9.0.3 is equivalent to Conjecture 9.0.2 in the case of Jacobians of genus two
curves.
Remark 9.0.4. For the quadratic points 𝑃 with (𝑃, 𝜎𝑃) = (𝑃, 𝜏𝑃) (the quadratic points coming
from pulling back 𝑘-rational points via the hyperelliptic map 𝜓 : 𝐶 → P1 ), the inequality of
Conjecture 9.0.3 does not hold. In fact, we can show that for such points
                                (6 − 𝜖)ℎ(𝑃) ≤ 𝑑 𝑎 (𝑃) ≤ (6 + 𝜖)ℎ(𝑃).
Consider 𝜓 × 𝜓 : 𝐶 × 𝐶 → P1 × P1 . Let 𝐹1 and 𝐹2 be fibers of the two natural projections on
P1 ×P1 . Indeed, since ℎ 𝜙∗ Θ (𝑃, 𝜎𝑃) = ℎΘ (0) is constant for such points, this follows immediately
from (9.1).
                                                  65


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