Hyperexponential solutions of elliptic difference equations

TL;DR

This paper introduces an algorithmic approach to find rational, pseudo-rational, and hyperexponential solutions of elliptic difference equations defined on elliptic curves with coefficients in a number field.

Contribution

It presents a novel algorithm for computing hyperexponential solutions of elliptic difference equations, expanding the solution space beyond rational functions.

Findings

Algorithm successfully computes rational solutions.

Extension to pseudo-rational solutions demonstrated.

Hyperexponential solutions characterized and computed.

Abstract

Consider an elliptic curve $\mathcal{C}$ with coefficients in $\mathbb{K}$ with $[\mathbb{K}:\mathbb{Q}]<\infty$ and $\delta \in \mathcal{C}(\mathbb{K})$ a non torsion point. We consider an elliptic difference equation $\sum_{i=0}^l a_i(p) f(p\oplus i.\delta)=0$ with $\oplus$ the elliptic addition law and $a_i$ polynomials on $\mathcal{C}$. We present an algorithm to compute rational solutions, then an intermediary class we call pseudo-rational solutions, and finally hyperexponential solutions, which are functions $f$ such that $f(p\oplus \delta)/f(p)$ is rational over $\mathcal{C}$.