Investigation into the role of the Laurent property in integrability

TL;DR

This paper explores the Laurent property in discrete equations, providing new proofs and conditions for its preservation, and demonstrating zero algebraic entropy in certain reductions, advancing understanding of integrability.

Contribution

It offers a novel proof of the Laurent property for key equations and establishes explicit conditions for nonautonomous cases, linking Laurent property to integrability.

Findings

Laurent property holds for Hirota-Miwa and discrete BKP equations without caterpillar lemma

Reductions and gauge transformations preserve the Laurent property

Reductions of equations with Laurent property have zero algebraic entropy

Abstract

We study the Laurent property for autonomous and nonautonomous discrete equations. First we show, without relying on the caterpillar lemma, the Laurent property for the Hirota-Miwa and the discrete BKP equations. Next we introduce the notion of reductions and gauge transformations for discrete bilinear equations and we prove that these preserve the Laurent property. Using these two techniques, we obtain the explicit condition on the coefficients of a nonautonomous discrete bilinear equation for it to possess the Laurent property. Finally we study the denominators of the iterates of an equation with the Laurent property and we show that any reduction to a mapping on a one-dimensional lattice of a nonautonomous Hirota-Miwa equation or discrete BKP equation, with the Laurent property, has zero algebraic entropy.