Algebraic entropy of an extended Hietarinta-Viallet equation

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

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.

Singularity confinement and chaos in two-dimensional discrete systems

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its iterates exhibit exponential growth. By systematic reduction to one-dimensional systems, it gives a hierarchy of ordinary difference equations with confined singularities, but with positive algebraic entropy including a generalized form of the Hietarinta-Viallet mapping. We believe that this is the first example of such quasi-integrable equations defined ov…

The redemption of singularity confinement

We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby…

任意维格子上保持互质性的离散 KdV 型方程

Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice

本文提出一个定义在多维格子上的方程，作为保持互质性的类离散KdV方程的扩展及Hietarinta-Viallet方程的高维类似物，主要证明了其tau-function形式的Laurent性质和不可约性质，从而推导出方程的互质性。

Hietarinta Yang-Baxter方程解：可逆4×4算符的代数分类

Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators

本文利用配分代数、Clifford代数、Temperley-Lieb代数及对易算符集合等代数拟设，以不依赖表示的方式分类常数Yang-Baxter解，复现了Hietarinta 11族可逆4×4解中的10族，为量子计算机模拟可积系统奠定基础。

A non autonomous generalization of the $Q_{\text{V}}$ equation

In this paper we introduce a non autonomous generalization of the $\QV$ equation introduced by Viallet. All the equations of Boll's classification appear in it for special choices of the parameters. Using the algebraic entropy test we infer that the equation should be integrable and with the aid of a formula introduced by Xenitidis we find its three point generalized symmetries.