Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

Abstract

International audienceLet L be a field of characteristic p with q elements and F ∈ L[X, Y ] be a polynomial with p > deg Y (F) and total degree d. In [40], we showed that rational Puiseux series of F above X = 0 could be computed with an expected number of O˜d 3 log q) arithmetic operations in L. In this paper, we reduce this bound to O˜og q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y ] with Oõperations in L. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

“…Consider the ψ g -adic expansions c i ψ i g+1 = j a j ψ j g and F = j α j ψ j g . Thanks to (24), there exists at least one index j such that (j, v g (a j ψ j g )) ∈ N g (c i ψ i g+1 ). By (24), N g (F ) is the lower convex hull of (j, v g (α j ψ j g )), which is by assumption straight of slope −q g+1 /m g+1 .…”

“…Thanks to (24), there exists at least one index j such that (j, v g (a j ψ j g )) ∈ N g (c i ψ i g+1 ). By (24), N g (F ) is the lower convex hull of (j, v g (α j ψ j g )), which is by assumption straight of slope −q g+1 /m g+1 . It follows that min j (q g+1 v g (a j ψ j g ) + m g+1 j) ≤ min j (q g+1 v g (α j ψ j g ) + m g+1 j).…”

“…Organisation. In Section 2 below, we describe briefly the rational Newton-Puiseux algorithm of Duval [6] and its improved version of [24] due to the so-called Abhyankar trick. In section 3, we show how to recover the edge data of F from its Φ-adic expansion, where Φ is the collection of minimal polynomials of the truncated Puiseux series of F .…”

“…The product of these invariants over all iterations satisfies k q k ℓ k ≤ d (with equality if and only if F is irreducible), and the number of calls is less than log(d). See [24,Section 4] for details. If F is irreducible, we have moreover v x (F y (S)) = δ d for any Puiseux series S of F .…”

Computing Puiseux series : a fast divide and conquer algorithm

“…With our assumption on p, the Puiseux theorem states that for any x 0 ∈ K, the roots of F (viewed as a univariate polynomial in Y ) may be expressed as fractional Laurent power series in (X − x 0 ) with coefficients in K. These are the (classical) Puiseux series 1 of F above x 0 , fundamental objects of the theory of algebraic curves [8,39]. Many applications are given in [32,33].…”