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Algebraic entropy for systems of quad equations

Giorgio Gubbiotti

Abstract

In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.

Algebraic entropy for systems of quad equations

Abstract

In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.
Paper Structure (13 sections, 2 theorems, 52 equations, 5 figures)

This paper contains 13 sections, 2 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. My relationship with Decio Levi
  3. Statement of the problem and outline of the paper
  4. Algebraic entropy for systems of quad equations with staircase initial conditions
  5. Examples
  6. Coupled lpKdV systems
  7. Lattice NLS system
  8. The lSG2 and the lmKdV2 systems
  9. Boussinesq-type systems
  10. Boussinesq system
  11. Schwarzian Boussinesq system
  12. Modified Boussinesq system
  13. Discussion and perspectives

Key Result

Theorem 2.6

A sequence $\left\{ d_{k} \right\}_{k\in\mathbb{N}_{0}}$ admits a rational generating function $g\in\mathbb{C}\left( s \right)$ if and only if it solves a linear difference equation with constant coefficients. Moreover, if $\rho>0$ is the radius of convergence of $g$, writing $g$ as: where $A$ and $B$ are analytic functions for $\abs{s}<r$ such that $B(\rho)\neq 0$ we have: where $\Gamma(s)$ is

Figures (5)

  • Figure 1: A quad-graph.
  • Figure 2: Regular and non-regular staircases.
  • Figure 3: Varius kinds of restricted initial conditions.
  • Figure 4: The range for the initial conditions $\Delta_{[-2,1]}^{(3)}$.
  • Figure 5: The four principal diagonals.

Theorems & Definitions (6)

  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Corollary 2.7