Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

TL;DR

This paper introduces a novel non-integrable extension of the two-dimensional Toda lattice that preserves coprimeness, demonstrating irreducibility and co-primality of iterates, and extends to lower-dimensional cases.

Contribution

It presents the first example of a coprimeness-preserving non-integrable discrete equation on a three-dimensional lattice, with proofs of irreducibility and co-primality.

Findings

All iterates are irreducible Laurent polynomials.

Every pair of iterates is co-prime.

The extension reduces to known non-integrable cases in lower dimensions.

Abstract

We introduce a so-called `coprimeness-preserving non-integrable' extension (another terminology is `quasi-integrable' extension) to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equation defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.