Quadratic Diophantine equations, the Heisenberg group and formal languages

TL;DR

This paper demonstrates that solutions to quadratic equations in integers and one-variable equations in the Heisenberg group can be described using EDT0L formal languages, revealing a connection between algebraic equations and formal language theory.

Contribution

It introduces a novel approach to describing solutions of algebraic equations in groups using formal languages, bridging group theory and formal language theory.

Findings

Solutions to quadratic equations in integers are expressible via EDT0L languages.

Solutions to one-variable equations in the Heisenberg group are also describable by EDT0L languages.

The reduction from group equations to integer equations leverages the link between nilpotent groups and rings of integers.

Abstract

We express the solutions to quadratic equations with two variables in the ring of integers using EDT0L languages. We use this to show that EDT0L languages can be used to describe the solutions to one-variable equations in the Heisenberg group. This is done by reducing the question of solving a one-variable equation in the Heisenberg group to solving an equation in the ring of integers, exploiting the strong link between the ring of integers and nilpotent groups.