Integrability criterion in terms of coprime property for the discrete Toda equation

Abstract

We reformulate the singularity confinement, which is one of the most famous integrability criteria for discrete equations, in terms of the algebraic properties of the general terms of the discrete Toda equation. We show that the coprime property, which has been introduced in our previous paper as one of the integrability criteria, is appropriately formulated and proved for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same coprime property holds for all the types of boundaries.