Differential equations for algebraic functions

TLDR

Univariate algebraic functions satisfy linear differential equations with polynomial coefficients, and their power series coefficients obey linear recurrences. The authors analyze the computational complexity of deriving these differential equations and recurrences, leading to a fast algorithm for expanding algebraic series. They prove that the minimal‑order differential equation has coefficients of cubic degree in the function’s degree, that a linear‑order equation exists with quadratic‑degree coefficients, and that near‑optimal recurrences in order and degree can be achieved.

Abstract

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series.