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Journals and Conferences
Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal… (More)
We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set of points obtained from the intersection of the component… (More)
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there… (More)
Let ƒ<subscrpt>1</subscrpt>, … , ƒ<subscrpt><italic>s</italic></subscrpt> be polynomials in <italic>n</italic> variables over a field of characteristic zero and <italic>d</italic> be the maximum of their total degree. We propose a new probabilistic algorithm for computing a <italic>geometric resolution</italic> of each equidimensional part… (More)
UNLABELLED BACKGROUND OBJECTIVE: Femoral offset is supposed to influence the results of hip replacement but little is known about the accurate method of measure and the true effect of offset modifications. MATERIAL AND METHODS This article is a collection of independent anatomic, radiological and clinical works, which purpose is to assess knowledge of the… (More)
We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in sub-quadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the… (More)
Current implementations of p-adic numbers usually rely on so called zealous algorithms, which compute with truncated p-adic expansions at a precision that can be specified by the user. In combination with Newton-Hensel type lifting techniques, zealous algorithms can be made very efficient from an asymptotic point of view. In the similar context of formal… (More)
We present new faster deterministic and probabilistic recombination algorithms to compute the irreducible decomposition of a bivariate polynomial via the classical Hensel lifting technique. For the dense bi-degree polynomial representation, the costs of our recombination algorithms are essentially sub-quadratic.
In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.