Investigation into the role of the Laurent property in integrability

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 (or so-called Miwa) 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.