Greatest common divisors and Vojta's conjecture for blowups of algebraic tori
In the paper under review, the author gives an inequality bounding the greatest common divisor of multivariate polynomials evaluated at \({\mathcal S}\)-unit arguments. The author's result extends the well-known result of Bugeaud, Corvaja and Zannier [\textit{Y. Bugeaud} et al., Math. Z. 243, No. 1, 79--84 (2003; Zbl 1021.11001)] on the greatest common divisor of \(a^n-1\) and \(b^n-1\) for multiplicatively independent integers \(a>1\) and \(b>1\) as well as its generalizations by Corvaja and Zanier, Fuchs, and Hernández and the reviewer. As related results, following ideas of Silverman, the author also proves special cases of Volta's conjectures for blow-ups of toric varieties. Finally, the author classifies when members of linear recurrent sequences with simple roots may have a ``large'' greatest common divisor infinitely often. The main tool is Schmidt's subspace theorem from Diophantine approximations.
scientific article; zbMATH DE number 7023767
scientific article