Abstract
Let F∈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 ˜(Dδ ) 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 ˜(D(δ +N)) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜(D 3 ) arithmetic operations and, if 𝕂=ℚ, with Ø ˜((h+1)D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F.
Highlights
This paper provides complexity results for computing Puiseux series of a bivariate polynomial with coefficients over a perfect field of characteristic zero or big enough
In [PR15], still considering K = Fpc, an algorithm is given to compute the singular part of Puiseux series over x0 = 0 in an expected O (ρ dY δ + ρ dY log(pc)) arithmetic operations, where ρ is the number of rational Puiseux expansions above x0 = 0
Remark 3.12. — [L19, Section 4] provides an almost linear deterministic algorithm to compute modulo tower of fields by computing “accelerated towers” instead of primitive elements. Such a strategy would lead to a version of Theorem 3.2 with a deterministic algorithm and a complexity bound O(dY 1+o(1) δ). [L20] deals with dynamic evaluation, so that this bound should propagate for our main results
Summary
This paper provides complexity results for computing Puiseux series of a bivariate polynomial with coefficients over a perfect field of characteristic zero or big enough. In [PR15], still considering K = Fpc, an algorithm is given to compute the singular part of Puiseux series over x0 = 0 in an expected O (ρ dY δ + ρ dY log(pc)) arithmetic operations, where ρ is the number of rational Puiseux expansions above x0 = 0 (bounded by dY ) These two algorithms use univariate factorisation over finite fields, cannot be directly extended to the zero characteristic case. This explains why the second result does not provide an improved bound for the computation of Puiseux series above all critical points. None of these methods have been proved to provide a complexity which fits in the bounds obtained in this paper
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