2007/03/31 by Alin Bostan, Frédéric Chyzak, Bruno Salvy +2 · 1 citation
Computer Science · Mathematics · #cs.SC #math.CA
paper · pdf · doi:10.1145/1277548.1277553
published as ISSAC'07, pages 25--32, ACM Press, 2007.
arxiv created 2007/09/14 · arxiv updated 2009/12/01
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.