Algebraic entropy of an extended Hietarinta-Viallet equation

TL;DR

This paper extends the Hietarinta-Viallet mapping with a parameter, analyzing its algebraic entropy through degree recurrence relations, and explores the role of singularity confinement, irreducibility, and co-primeness.

Contribution

It introduces a parameterized extension of the Hietarinta-Viallet mapping and derives algebraic entropy using recurrence relations for degrees.

Findings

For some parameters, the mapping has confined singularities.

The mapping's algebraic entropy is obtained via degree recurrence relations.

Irreducibility and co-primeness are key in analyzing the mapping's properties.

Abstract

We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping. For some parameter values the mapping has a confined singularity, in which case the mapping is equivalent to a recurrence relation between irreducible polynomials. For other parameter values, the mapping does not pass the singularity confinement test. The properties of irreducibility and co-primeness of the terms play crucial roles in the discussion.