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 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.

References in corpus (4)

• Singularity confinement for maps with the Laurent property
• Bilinear equations and $q$-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras
• Irreducibility and co-primeness as an integrability criterion for discrete equations
• Algebraic entropy of an extended Hietarinta-Viallet equation

Cited by in corpus (18)

• Some integrable maps and their Hirota bilinear forms
• On Reductions of the Hirota-Miwa Equation
• On the singularity structure of the discrete KdV equation
• Full-deautonomisation of a lattice equation
• Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation
• On the Coprimeness Property of Discrete Systems without the Irreducibility Condition
• Mutations of the cluster algebra of type ${\mathbf{A}}_1^{(1)}$ and the periodic discrete Toda lattice
• Degree growth of lattice equations defined on a 3x3 stencil
• Singularities and growth of higher order discrete equations
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations
• Linear relations for Laurent polynomials and lattice equations
• Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type
• A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
• Super-QRT and 4D-mappings reduced from the lattice super-KdV equation
• QRT maps and related Laurent systems
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
• Periodicity, linearizability and integrability in seed mutations of type $A_N^{(1)}$