Differential Equations for Algebraic Functions

TL;DR

This paper investigates the differential equations satisfied by algebraic functions, establishing bounds on their order and degree, and presents efficient algorithms for computing series expansions.

Contribution

It provides new bounds on the order and degree of minimal differential equations for algebraic functions and introduces a fast algorithm for algebraic series expansion.

Findings

Minimal order differential equation has coefficients with degree cubic in the function's degree.

Existence of a linear order differential equation with quadratic degree coefficients.

Development of a fast algorithm for algebraic series expansion.

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.