Families of Unit Equations and Exponential Diophantine Problems via Integral Points

TL;DR

This paper explores integral points on projective varieties using advanced theorems, proving degeneracy results for unit equations and extending GCD estimates to exponential Diophantine equations.

Contribution

It introduces a higher-dimensional generalization of Huang-Levin-Xiao inequalities and applies it to new degeneracy results and GCD estimates in exponential Diophantine problems.

Findings

Degeneracy results for solutions of unit equations with polynomial coefficients.

Extension of GCD estimates to specific exponential Diophantine equations.

Insights into digit distribution in $q$-adic representations.

Abstract

This paper investigates the distribution of integral points on projective varieties via two distinct methods: the Ru-Vojta theorem and our higher-dimensional generalization of the Huang-Levin-Xiao inequalities. These approaches operate under distinct geometric conditions, specifically the transverse and proper intersections of boundary divisors. Applying this framework, we prove degeneracy results for the solution sets of two classes of one-parameter families of unit equations, differentiated by the degrees of their polynomial coefficients. Finally, we extend previous greatest common divisor (GCD) estimates to derive new results for specific exponential Diophantine equations and the distribution of digits in $q$-adic representations.