Results 1  10
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30
Computing Parametric Geometric Resolutions
, 2001
"... Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we f ..."
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Cited by 29 (8 self)
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Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 22 (10 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Fast algorithms for zerodimensional polynomial systems using duality
 APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
, 2001
"... Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the q ..."
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Cited by 17 (3 self)
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Many questions concerning a zerodimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A = k[X1,..., Xn]/I, where I is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the Amodule structure of the dual space � A. An important feature of our algorithms is that we do not require � A to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 n D 5/2) and O(n2 n D 5/2) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.
POINT COUNTING IN FAMILIES OF HYPERELLIPTIC CURVES IN CHARACTERISTIC 2
"... Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations ..."
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Cited by 14 (5 self)
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Let ĒΓ be a family of hyperelliptic curves over F2 alg cl with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ē¯γ, with ¯γ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and memory O(n2.5), and the computation for n curves of the family can be done in time Õ(n3.376). All of these algorithms are polynomialtime in the genus.
Fast computation with two algebraic numbers
 September
, 2002
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 7 (3 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
Computing the eigenvalue in the SchoofElkiesAtkin algorithm using Abelian lifts
"... The SchoofElkiesAtkin algorithm is the best known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. We use Abelian properties of division polynomials to design a fast theoretical and practical algorithm for finding the eigenvalue. 1. ..."
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Cited by 6 (2 self)
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The SchoofElkiesAtkin algorithm is the best known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. We use Abelian properties of division polynomials to design a fast theoretical and practical algorithm for finding the eigenvalue. 1.
Fast algorithms for computing the eigenvalue in the SchoofElkiesAtkin algorithm
 In ISSAC’06
, 2006
"... The SchoofElkiesAtkin algorithm is the only known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. Several practical and asymptotical improvements for the phase called eigenvalue computation are proposed. 1. ..."
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Cited by 4 (2 self)
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The SchoofElkiesAtkin algorithm is the only known method for counting the number of points of an elliptic curve defined over a finite field of large characteristic. Several practical and asymptotical improvements for the phase called eigenvalue computation are proposed. 1.