Separated Variables on Plane Algebraic Curves
Manfred Buchacher
TL;DR
The paper tackles the problem of determining whether the restriction of a rational function $r$ to the plane curve defined by an irreducible polynomial $p$ can be written as $f(x) - g(y)$ with $f \in \mathbb{K}(x)$ and $g \in \mathbb{K}(y)$. It advances this question by developing a complete algorithm in the case where the field of separated multiples $\mathrm{F}(p)$ is non-trivial, and by formulating conjectural semi-algorithms for the case $\mathrm{F}(p) \cong \mathbb{K}$, both relying on a detailed pole/multiplicity analysis via Puiseux expansions and orbit graphs on the curve. The work connects elimination problems in algebraic dynamics and the construction of invariants/decoupling functions with applications to linear discrete differential equations and lattice-walk enumerations, and it outlines several open problems and conjectures about orbit finiteness and algorithm termination. Overall, it provides a framework for reducing nonlinear separation problems to linear systems and highlights the role of curve dynamics in rational function decomposition on algebraic curves.
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.