Computing Puiseux series: a fast divide and conquer algorithm

Poteaux, Adrien; Weimann, Martin

doi:10.5802/ahl.97

Poteaux, Adrien; Weimann, Martin

Computing Puiseux series: a fast divide and conquer algorithm

Annales Henri Lebesgue, Volume 4 (2021), pp. 1061-1102.

Metadata

DOI10.5802/ahl.97

Keywords Puiseux series, complexity, dynamic evaluation

Abstract

Let $F \in 𝕂 [X, Y]$ be a polynomial of total degree $D$ defined over a perfect field $𝕂$ of characteristic zero or greater than $D$ . Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes all singular parts of Puiseux series of $F$ above $X = 0$ in an expected $Ø \tilde{} (D δ_{})$ operations in $𝕂$ , where $δ_{}$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$ . To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $𝕂 [[X]] [Y]$ up to an arbitrary precision $X^{N}$ with $Ø \tilde{} (D (δ_{} + N))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $Ø \tilde{} (D^{3})$ arithmetic operations and, if $𝕂 = ℚ$ , with $Ø \tilde{} ((h + 1) D^{3})$ bit operations using probabilistic algorithms, where $h$ is the logarithmic height of $F$ .

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