Differential Equations for Algebraic Functions

Details Differential Equations for Algebraic Functions Differential Equations for Algebraic Functions

2007-03-23

arxiv.org/abs/cs/0703121v2

ISSAC'07, pages 25--32, ACM Press, 2007.

Computer Science

Symbolic Computation

Scientific paper

10.1145/1277548.1277553

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.

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