Irreducibility and co-primeness as an integrability criterion for discrete equations

TL;DR

This paper demonstrates that irreducibility and co-primeness of terms serve as effective criteria to distinguish integrable discrete equations from non-integrable ones, extending previous singularity confinement concepts.

Contribution

It establishes co-primeness as a new mathematical criterion for integrability, generalizing earlier results to broader initial and boundary conditions.

Findings

Irreducibility and co-primeness hold only in integrable cases.

Co-primeness can be used as an integrability criterion.

Results extend to general initial and boundary conditions.

Abstract

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg-de Vries (dKdV) equation. In our previous paper (arXiv:1311.0060), we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations. v2, v3: minor revisions