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14
Fast arithmetic for triangular sets: from theory to practice
 ISSAC'07
, 2007
"... We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, ..."
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Cited by 32 (24 self)
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We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasilinear complexity. The main outcome we have in mind is the acceleration of higherlevel algorithms, by interfacing our lowlevel implementation with languages such as AXIOM or Maple. We show the potential for huge speedups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.
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.
Improved dense multivariate polynomial factorization algorithms
 J. Symbolic Comput
, 2005
"... We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the numb ..."
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Cited by 19 (3 self)
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We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several to one variable. The deterministic algorithm runs in subquadratic 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 reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem. Key words: Polynomial factorization, Hensel lifting, Bertini’s irreducibility theorem.
Homotopy techniques for multiplication modulo triangular sets
, 2009
"... We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluationinterpolation techniques. We obtain a quasilinear time complexity for substantial families of exa ..."
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We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluationinterpolation techniques. We obtain a quasilinear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of algebraic numbers in small characteristic.
Xavier Caruso
"... We design an algorithm for computing the pcurvature of a differential system in positive characteristic p. For a system of dimension r with coefficients of degree at most d, its complexity is O (̃pdrω) operations in the ground field (where ω denotes the exponent of matrix multiplication), whereas ..."
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We design an algorithm for computing the pcurvature of a differential system in positive characteristic p. For a system of dimension r with coefficients of degree at most d, its complexity is O (̃pdrω) operations in the ground field (where ω denotes the exponent of matrix multiplication), whereas the size of the output is about pdr2. Our algorithm is then quasioptimal assuming that matrix multiplication is (i.e. ω = 2). The main theoretical input we are using is the existence of a wellsuited ring of series with divided powers for which an analogue of the Cauchy–Lipschitz Theorem holds. Categories and Subject Descriptors: