Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type

TL;DR

This paper extends the Hietarinta-Viallet equation to a multi-term recurrence, analyzing its algebraic entropy and demonstrating nonintegrability despite coprimeness and irreducibility.

Contribution

It introduces a new family of recurrences derived from the discrete KdV equation and calculates their algebraic entropy using an elementary degree growth method.

Findings

Recurrence satisfies irreducibility and coprimeness properties.

The algebraic entropy is explicitly derived.

The recurrence is shown to be nonintegrable despite these properties.

Abstract

We introduce a family of extensions of the Hietarinta-Viallet equation to a multi-term recurrence relation via a reduction from the coprimeness-preserving extension to the discrete KdV equation. The recurrence satisfies the irreducibility and the coprimeness property although it is nonintegrable in terms of an exponential degree growth. We derive the algebraic entropy of the recurrence by an elementary method of calculating the degree growth. The result includes the entropy of the original Hietarinta-Viallet equation.