Groups with context-free Diophantine problem

TL;DR

This paper characterizes groups based on the complexity of their Diophantine problem, showing that only finite groups have a context-free Diophantine problem, linking algebraic properties with formal language theory.

Contribution

It establishes a precise algebraic condition connecting the finiteness of a group to the context-freeness of its Diophantine problem, a novel intersection of group theory and formal languages.

Findings

Finite groups have context-free Diophantine problems.

Infinite groups do not have context-free Diophantine problems.

Provides algebraic criteria for the complexity classification of Diophantine problems.

Abstract

We find algebraic conditions on a group equivalent to the position of its Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a finitely generated group has a context-free Diophantine problem if and only if it is finite.