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)
The transposition principle, also called Tellegen's principle, is a set of transformation rules for linear programs. Yet, though well known, it is not used systematically, and few practical implementations rely on it. In this article, we propose explicit transposed versions of polynomial multiplication and division but also new faster algorithms for… (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)
We present new faster deterministic and probabilistic recombina-tion 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.
Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision <i>linear</i> in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using <i>trace recombination</i>. Then, the total cost of the… (More)
In the vein of recent algorithmic advances in polynomial factor-ization 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.
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)
We give a new proof of Fürer's bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike Fürer, our method does not require constructing special coecient rings with fast roots of unity. Moreover, we prove the more explicit bound O(n logn K log n) with K = 8. We show that an optimised variant of Fürer's algorithm achieves only K =… (More)