On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

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.