Greatest common divisors near S-units, applications, and conjectures on arithmetic abelian surfaces

We bound the greatest common divisor of two coprime multivariable polynomials evaluated at algebraic numbers, generalizing the work of Levin by thickening the finitely generated group to allow non-finitely generated elements. Going towards conjectured inequalities of Silverman and Vojta, an immediate corollary shows a similar inequality without a normal crossing assumption. 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. The exceptional set is not as good as finitely many linear relations as in the simple case, but within the control of a logarithmic region, improving recent results of Grieve and Wang. An example shows that the logarithmic region is necessary.On abelian surfaces which come from the jacobians of hyperelliptic curves, we establish the connection between the GCD conjecture and the conjecture on arithmetic discriminant. It predicts, under particular situations, stronger inequality than Vojta's theorem of the arithmetic discriminant. We give some examples of extreme values of the arithmetic discriminant.