International audienc

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

International audienceWe have designed a new symbolic-numeric strategy for computing efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well-chosen prime number p are used to obtain the exact information needed to guide floating point computations. In this paper, we detail the symbolic part of our algorithm. First of all, we study modular reduction of Puiseux series and give a good reduction criterion for ensuring that the information required by the numerical part is preserved. To establish our results, we introduce a simple modification of classical Newton polygons, that we call "generic Newton polygons", which turns out to be very convenient. Finally, we estimate the size of good primes obtained with deterministic and probabilistic strategies. Some of these results were announced without proof at ISSAC'08