Integrability criterion in terms of coprime property for the discrete Toda equation

TL;DR

This paper establishes a co-primeness integrability criterion for the discrete Toda equation, demonstrating its validity across various boundary conditions, thus advancing the understanding of discrete integrable systems.

Contribution

It proves the co-primeness property as an integrability criterion for the discrete Toda equation under multiple boundary conditions.

Findings

Co-primeness property holds for all boundary types studied.

Singularity confinement is reformulated in terms of co-primeness.

The results support co-primeness as a robust integrability criterion.

Abstract

We reformulate the singularity confinement of the discrete Toda equation. We prove the co-primeness property, which has been introduced in our previous paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same co-primeness property holds for all the types of boundaries. (v2: typos corrected, final version to appear in J. Math. Phys.)