Investigation into the role of the Laurent property in integrability

Abstract

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.

Some integrable maps and their Hirota bilinear forms

“…So the iterates of (10) coincide with those of (5), subject to fixing γ 0 , γ 1 as above. Now we can make use of proposition 5.4 in [30], which implies that the nonautonomous Somos recurrence (5) has the Laurent property, meaning that…”

“…In the context of integrability, the Laurent property appears at the level of Hirota bilinear equations: the Hirota-Miwa (discrete KP) equation can be derived from mutations in a cluster algebra [32], which means that it has the Laurent property, and this property is inherited by its reductions to recurrences of Somos (or Gale-Robinson) type [5,30]. Furthermore, it seems likely that any birational map with confined singularities can be lifted to a higher-dimensional 'Laurentified' system, i.e.…”

Cluster algebras and discrete integrability

“…Both case (ii), which corresponds to affine quivers of typeà q,N −q , and case (iii) display linear degree growth, similar to Example 9. Case (iv) consists of Somos-N recurrences, which display quadratic degree growth [56], as in Example 11. Hence only zero, linear, quadratic or exponential growth is displayed by the cluster recurrences (19).…”

Mutations of the cluster algebra of type ${A}_{1}^{(1)}$ and the periodic discrete Toda lattice

“…Since their introduction cluster algebras have found applications in the field of dynamical systems such as Y -systems, discrete soliton equations and discrete Painlevé equations [3,8,9,10,12,15,13]. In this paper, we proceed to study links of cluster algebras and discrete integrable systems and establish a direct connection between seed mutations in the cluster algebra of type A (1) 1 and time evolutions of the periodic discrete Toda lattice of the lowest dimension through their identifications with QRT maps.…”

“…Then χ 1 generates a sequences of points starting from I 0 , V 0 on C 4 since (x 0 , y 0 ) is not a base point of the pencil { γ} λ∈P 1 (C) (see (13)). Thus, we identify the map χ 1 with the map ϕ TL .…”