A study on the domain independence of the Laurent property, the irreducibility and the coprimeness in lattice equations

TL;DR

This paper investigates how the Laurent property, irreducibility, and coprimeness in lattice equations are intrinsic to the equations themselves, regardless of initial conditions or domain choices, and explores their preservation under reductions.

Contribution

It proves that these properties are domain-independent and inherent to the equations, and demonstrates their preservation during reductions, even with torsion elements.

Findings

Properties are independent of initial domain choices.

Reductions preserve the Laurent property.

Properties hold even with torsion elements in lattices.

Abstract

We study the Laurent property, the irreducibility and the coprimeness for lattice equations (partial difference equations), mainly focusing on how the choice of initial value problem (the choice of domain) affects these properties. We show that these properties do not depend on the choice of domain as long as they are considered together. In other words, these properties are inherent to a difference equation. Applying our result, we discuss the reductions of lattice equations. We show that any reduction of a Laurent system, even if the lattices have torsion elements, preserves the Laurent property.