Effective results for unit equations over finitely generated domains

TL;DR

This paper proves an effective upper bound on solutions to unit equations over finitely generated domains, extending previous results to more general algebraic structures and providing explicit size estimates.

Contribution

It extends effective finiteness results for unit equations to arbitrary finitely generated domains over Z, with explicit bounds on solutions.

Findings

Provides explicit bounds for solutions in finitely generated domains.

Extends effective results beyond number fields to general finitely generated domains.

Utilizes existing effective results and specialization techniques for the proof.

Abstract

Let A be a commutative domain containing Z which is finitely generated as a Z-algebra, and let a,b,c be non-zero elements of A. It follows from work of Siegel, Mahler, Parry and Lang that the equation (*) ax+by=c has only finitely many solutions in elements x,y of the unit group A* of A, but the proof following from their arguments is ineffective. Using linear forms in logarithms estimates of Baker and Coates, in 1979 Gy\H{o}ry gave an effective proof of this finiteness result, in the special case that A is the ring of S-integers of an algebraic number field. Some years later, Gy\H{o}ry extended this to a restricted class of finitely generated domains A, containing transcendental elements. In the present paper, we give an effective finiteness proof for the number of solutions of (*) for arbitrary domains A finitely generated over Z. In fact, we give an explicit upper bound for the `sizes' of the solutions x,y, in terms of defining parameters for A,a,b,c. In our proof, we use already existing effective finiteness results for two variable S-unit equations over number fields due to Gy\H{o}ry and Yu and over function fields due to Mason, as well as an explicit specialization argument.