Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypertranscendence and linear difference equations, the exponential case

Thomas Dreyfus

TL;DR

The paper studies meromorphic solutions $f$ to linear shift difference equations with coefficients in $\mathbb{C}(x)$ under the operator $\rho(y)=y(x+h)$. Using the parametrized difference Galois theory, it shows that if a solution vector to $\rho(Y)=AY$ is differential algebraic over $\mathbb{C}(x)$, then each coordinate lies in the ring generated by $h$-periodic functions and exponentials, i.e., $f_i\in C_{\,\ell h}(x)[e^{\lambda_1 x},\dots,e^{\lambda_k x}]$ for some $\ell$ and constants $\lambda_j$. The analysis blends difference Galois theory with $(\rho,\delta)$-PV theory to handle the interplay between shift and differentiation, including irreducible Galois groups and affine reductions. The results generalize known hypertranscendence phenomena to the shift setting and identify precise structural forms for differentially algebraic solutions. This provides a sharp dichotomy: either the solutions are extremely simple (exponential-polynomial in $x$ with $h$-periodic factors) or they are differentially transcendental.

Abstract

In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in $\mathbb{C}(x)$ involving the operator $ρ: y(x)\mapsto y(x+h)$, for some $h\in \mathbb{C}^*$. We prove that if $f$ is solution of an algebraic differential equation, then $f$ belongs to a ring that is made with periodic functions and exponentials. Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer.

Hypertranscendence and linear difference equations, the exponential case

TL;DR

The paper studies meromorphic solutions to linear shift difference equations with coefficients in under the operator . Using the parametrized difference Galois theory, it shows that if a solution vector to is differential algebraic over , then each coordinate lies in the ring generated by -periodic functions and exponentials, i.e., for some and constants . The analysis blends difference Galois theory with -PV theory to handle the interplay between shift and differentiation, including irreducible Galois groups and affine reductions. The results generalize known hypertranscendence phenomena to the shift setting and identify precise structural forms for differentially algebraic solutions. This provides a sharp dichotomy: either the solutions are extremely simple (exponential-polynomial in with -periodic factors) or they are differentially transcendental.

Abstract

In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in involving the operator , for some . We prove that if is solution of an algebraic differential equation, then belongs to a ring that is made with periodic functions and exponentials. Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer.
Paper Structure (5 sections, 14 theorems, 18 equations)

This paper contains 5 sections, 14 theorems, 18 equations.

Table of Contents

  1. Aknowledgment
  2. Difference setting
  3. Differential setting
  4. Irreducible Galois group
  5. Proof of the main result

Key Result

Theorem 1

Let $n\in\mathbb{N}^*$, and let $A\in \mathrm{GL}_n (\mathbb{C}(x))$. Let $\mathcal{Y}:= (f_1,\dots, f_n)^{\top}\in (\mathcal{M}(\mathbb{C}))^n$ be a solution of $\rho(Y)=AY$. If every $f_i$ is differentially algebraic over $\mathbb{C}(x)$, then there exist $\ell\in \mathbb{N}^*$, $\lambda_{1}\dots,

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 2
  • proof : Proof of Corollary \ref{['cor']}
  • Example 3
  • Lemma 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 16 more