GCD inequalities inspired by Vojta’s conjecture

We prove several GCD inequalities inspired by Vojta’s deep conjecture in Diophantine geometry. These inequalities work with all algebraic numbers, so they can be thought of as attempts to extend GCD inequalities for S -units obtained by Corvaja and Zannier. Some of the inequalities we derive are weakened versions of inequalities obtained by Silverman under the assumption of Vojta’s conjecture. The upper bounds for the GCD’s in these inequalities all involve some contributions of heights and some contributions coming from the local heights outside S . The main ingredient of the proofs is the recent Diophantine approximation result of Ru and Vojta which is based on Schmidt subspace theorem and which involves a birational invariant coming from the asymptotic growth of the number of global sections of certain line bundles.