Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Systems of Discrete Differential Equations, Constructive Algebraicity of the Solutions

Hadrien Notarantonio, Sergey Yurkevich

TL;DR

A constructive and elementary proof of algebraicity of the solutions of such equations of order k with one catalytic variable is provided and an algorithm for computing annihilating polynomials of the algebraic series is provided.

Abstract

In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006) who settled down the case $n=1$. Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.

Systems of Discrete Differential Equations, Constructive Algebraicity of the Solutions

TL;DR

A constructive and elementary proof of algebraicity of the solutions of such equations of order k with one catalytic variable is provided and an algorithm for computing annihilating polynomials of the algebraic series is provided.

Abstract

In this article, we study systems of , not necessarily linear, discrete differential equations (DDEs) of order with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-Mélou and Jehanne (2006) who settled down the case . Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.
Paper Structure (24 sections, 13 theorems, 82 equations, 1 algorithm)

This paper contains 24 sections, 13 theorems, 82 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Main goals:
  4. Previous works
  5. Contributions
  6. Structure of the paper:
  7. Notations:
  8. General strategy and application to a first example
  9. Proofs of \ref{['thm:main_thm']} and \ref{['thm:quantitative_estimates']}
  10. Proof of \ref{['thm:main_thm']}
  11. Case $\mathbf{1}$:
  12. Case $\mathbf{2}$:
  13. Proof of \ref{['thm:quantitative_estimates']}
  14. Elementary strategies for solving systems of DDEs
  15. The classical duplication of variables algorithm
  16. ...and 9 more sections

Key Result

Theorem 1

Let $n, k\geq 1$ be integers and $f_1, \ldots, f_n\in\mathbb{K}[u]$, $Q_1, \ldots, Q_n\in\mathbb{K}[y_1, \ldots, y_{n(k+1)}, t, u]$ be polynomials. For $a\in\mathbb{K}$, set $\nabla_a^k F := (F, \Delta_a F, \ldots, \Delta_a^k F)$. Then the system of equations admits a unique vector of solutions $(F_1, \ldots, F_n)\in\mathbb{K}[u][[t]]^n$, and all its components are algebraic functions over $\math

Theorems & Definitions (27)

  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 17 more