Equations in Algebras

TL;DR

This paper proves that solving equations in free associative algebras and certain group algebras is undecidable, extending the scope of known undecidability results in algebraic structures.

Contribution

It establishes the undecidability of the Diophantine problem in free associative algebras and various group algebras over any field, broadening previous results.

Findings

Undecidability of equations in free associative algebras.

Undecidability in group algebras over torsion free groups.

Applicability to a wide class of algebraic structures.

Abstract

We show that the Diophantine problem(decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.