Algebraic entropy for face-centered quad equations

TL;DR

This paper introduces an algebraic entropy test for face-centered quad equations, demonstrating that equations satisfying a new consistency condition exhibit predictable growth patterns, aiding classification.

Contribution

It defines algebraic entropy for face-centered quad equations and applies it to CAFCC equations, revealing their growth behavior and integrability properties.

Findings

CAFCC equations pass the algebraic entropy test

Equations exhibit quadratic or linear growth

The test helps classify integrable face-centered quad equations

Abstract

In this paper we define the algebraic entropy test for face-centered quad equations, which are equations defined on vertices of a quadrilateral plus an additional interior vertex. This notion of algebraic entropy is applied to a recently introduced class of these equations that satisfy a new form of multidimensional consistency called consistency-around-a-face-centered-cube (CAFCC), whereby the system of equations is consistent on a face-centered cubic unit cell. It is found that for certain arrangements of equations (or pairs of equations) in the square lattice, all known CAFCC equations pass the algebraic entropy test possessing either quadratic or linear growth.