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Conference Papers Year : 2007

Differential Equations for Algebraic Functions

Alin Bostan
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Frédéric Chyzak
Bruno Salvy
Éric Schost
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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.
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Dates and versions

inria-00138206 , version 1 (23-03-2007)
inria-00138206 , version 2 (14-09-2007)

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Alin Bostan, Frédéric Chyzak, Bruno Salvy, Grégoire Lecerf, Éric Schost. Differential Equations for Algebraic Functions. ISSAC, Jul 2007, Waterloo, Canada. pp.8, ⟨10.1145/1277548.1277553⟩. ⟨inria-00138206v2⟩
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