Quadratic Diophantine equations, the Heisenberg group and formal languages

Alex Levine

doi:10.1007/s11856-025-2746-x

Quadratic Diophantine equations, the Heisenberg group and formal languages

Alex Levine

School of Engineering, Mathematics and Physics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	337-383
Number of pages	47
Journal	Israel Journal of Mathematics
Volume	269
Issue number	1
Early online date	27 Mar 2025
DOIs	https://doi.org/10.1007/s11856-025-2746-x
Publication status	Published - 1 Oct 2025

Keywords

quadratic Diophantine equations
equations in groups
EDT0L languages
nilpotent groups

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10.1007/s11856-025-2746-x

Quadratic Diophantine equations, the Heisenberg group and formal languagesAccepted author manuscript, 455 KBLicence: CC BY

Cite this

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AB - 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.

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KW - nilpotent groups

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Quadratic Diophantine equations, the Heisenberg group and formal languages

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Keywords

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