On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

TL;DR

This paper demonstrates that certain discrete equations maintain the coprimeness property even when their right-hand sides are factorizable, extending previous results that required irreducibility.

Contribution

It proves the coprimeness property for higher power extensions of discrete equations without requiring irreducibility of the right-hand side.

Findings

Coprimeness property holds despite factorizability

Irreducibility is not necessary for coprimeness

Extends understanding of discrete system properties

Abstract

In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.