# Toda type equations over multi-dimensional lattices

## TL;DR

This paper introduces a new class of higher-dimensional Toda lattice equations on multi-dimensional lattices, demonstrating their integrability properties through coprimeness and Laurent phenomenon algebra representations.

## Contribution

It extends the discrete Toda lattice to multi-dimensional lattices and proves their integrability via coprimeness and algebraic mutation properties.

## Key findings

- Equations satisfy the coprimeness property, indicating integrability.
- Iterate degrees grow exponentially, yet singularity behavior remains integrable.
- Equations can be expressed as mutations in Laurent phenomenon algebra.

## Abstract

We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.