Linear equations over multiplicative groups, recurrences, and mixing I

TL;DR

This paper develops effective methods for solving equations involving linear varieties and finitely generated multiplicative groups over fields of positive characteristic, with applications to S-unit equations.

Contribution

It introduces explicit height-based algorithms for computing intersections and solving S-unit equations in multiple variables over such fields.

Findings

Effective computation of intersections of linear varieties and multiplicative groups.

Explicit height estimates for solutions.

Solution of the S-unit equation in multiple variables.

Abstract

Let K be a field of positive characteristic. When V is a linear variety in K^n and G is a finitely generated subgroup of K^*, we show how to compute the intersection of V and G^n effectively using heights. We calculate all the estimates explicitly. A special case provides the effective solution of the S-unit equation in n variables.