Irreducibility and co-primeness as an integrability criterion for discrete equations

Abstract

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg-de Vries (dKdV) equation. In our previous paper [1] we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical reinterpretation of the confinement of singularities in the case of discrete equations.

Algebraic entropy of (integrable) lattice equations and their reductions

“…For ABS equations, parameters {α i , β j | i = 0, ..., 10, j = 0, ....10} are also taken randomly with integer values such that the corresponding equations are not degenerate. They confirm the agreement d l+1,m+1 = a l+1,m+1 where d l+1,m+1 is calculated from (16) with (8) and (9) and a l+1,m+1 from (26), with the corresponding same boundary values in each case. Consequently, we call a l+1,m+1 the ambient degree at the vertex (i.e.…”

A two-dimensional lattice equation as an extension of the Heideman–Hogan recurrence

“…The celebrated Hietarinta-Viallet equation is one of the mappings which have singularity confinement property but have positive algebraic entropy [2]. Recently, Mada and three of the authors have proposed a criterion, called the 'co-primeness property', which is regarded as an algebraic reinterpretation of the singularity confinement property [5]. We have constructed a two dimensional discrete lattice equation which has co-primeness property.…”

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

“…Here, r ≥ 2 and GCD(m, l, k) = 1. It is already obtained that the original Somos-4 has the Laurent property, the irreducibility and the coprimeness [5,15]. Each iterate z n of (4.…”

“…It is called the coprimeness property and is defined over the field of rational functions of the intial variables [16]. The coprimeness property is one type of singularity analysis of a discrete equation, which is quite similar to the singularity confinement test and is proved to be satisfied for many of the known discrete integrable systems [14,15]. The Laurent and the irreducibility properties played important roles in proving the coprimeness property of the given equations.…”