Solving algebraic equations in roots of unity

TL;DR

This paper advances the understanding of solutions to polynomial equations in roots of unity by providing explicit bounds and a constructive algorithm for identifying maximal torsion cosets on algebraic varieties.

Contribution

It introduces new explicit upper bounds for maximal torsion cosets and develops a constructive algorithm, improving upon previous polynomial growth bounds.

Findings

Derived new explicit upper bounds for maximal torsion cosets.

Developed a constructive algorithm for identifying torsion cosets.

Contrasted results with earlier polynomial growth bounds.

Abstract

This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic $n$-torus ${\mathbb G}_{\rm m}^n$. In contrast to earlier works that give the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of ${\mathbb G}_{\rm m}^n$.