Nonlinear forms of coprimeness preserving extensions to the Somos-$4$ recurrence and the two-dimensional Toda lattice equation --investigation into their extended Laurent properties--

TL;DR

This paper extends the coprimeness and Laurent properties to complex discrete systems like the Somos-4 recurrence and 2D Toda lattice, revealing their singularity structures and integrability features.

Contribution

It introduces a generalized coprimeness and Laurent property framework for non-autonomous discrete equations with complex singularities.

Findings

Extended Laurent property for non-autonomous forms

Proof of extended coprimeness property for studied equations

Insights into singularity structures of complex discrete systems

Abstract

Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-$4$ recurrence and the two-dimensional discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomous terms. Using this Laurent property, we prove that these equations satisfy the extended coprimeness property. This coprimeness property reflects the singularities that trivially arise from the equations.