Complexity bounds for the rational Newton-Puiseux algorithm over finite fields

Abstract

International audienc

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

“…This result should be compared with the bound O˜(d 5 + d 3 log q) given in [40], where we also derived an O˜(d 5 log q) bound for the computation of the genus of the curve defined by F and new complexity results for the number field case. Unfortunately, the improvement given by Theorem 1 does not propagate to genus computation; we will discuss this issue in the conclusion.…”

“…Merle and Henry [27], then Teitelbaum [46] studied the arithmetic complexity of the resolution of the singularity at the origin defined by F (X, Y ) = 0, a process tightly related to Puiseux series [9]. We have commented on these works and explained why we prefer to stick to the Newton-Puiseux algorithm in [40,41].…”

“…In [40], we studied how to truncate coefficients throughout the computations and gave a detailed count of the number of arithmetic operations in L required by Duval's version of the Newton-Puiseux algorithm to compute RPEs. Inspired by a proof by Abhyankar of Newton-Puiseux's Theorem [1,Chapter 12], we show herein that it is possible to improve this result using Hensel-like factorizations and simple, but essential, substitutions.…”

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Let 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 5 +d 3 log q) arithmetic operations in L. In this paper, we reduce this bound to O˜(d 4 +d 2 log 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.

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Polynomial root finding over local rings and application to error correcting codes

“…In general, even if F is separable, its residue polynomial modulo m may have multiple factors, and one has to make use of the Newton polygon technique recursively, assuming that the characteristic is sufficiently large. Over the power series, namely when R = ❑[[t]], several authors have contributed to this subject including, for instance: [13,14,16,27,28,38,46,47,48]. Over the p-adic integers the situation becomes more problematic but some of the latter techniques can be extended as in [27].…”