Separated Variables on Plane Algebraic Curves

TL;DR

This paper explores an algorithmic approach to determine when a rational function restricted to an algebraic curve can be expressed as a sum of functions in separate variables, with implications for algebraic geometry and rational function decomposition.

Contribution

It introduces an algorithm and a conjectural semi-algorithm for identifying when a restricted rational function can be separated into variables on algebraic curves.

Findings

Developed an algorithm for variable separation on algebraic curves

Proposed a semi-algorithm for cases with non-trivial rational multiples

Addressed the problem of expressing restricted functions as sums of single-variable functions

Abstract

We investigate the problem of deciding whether the restriction of a rational function $r\in\mathbb{K}(x,y)$ to the curve associated with an irreducible polynomial $p\in\mathbb{K}[x,y]$ is the restriction of an element of $\mathbb{K}(x)+\mathbb{K}(y)$. We present an algorithm and a conjectural semi-algorithm for finding such elements depending on whether $p$ has a non-trivial rational multiple in $\mathbb{K}(x) + \mathbb{K}(y)$ or not.