Algebraic entropy of (integrable) lattice equations and their reductions

Abstract

We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler–Bobenko–Suris equations and Viallet’s and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.