Homotopy techniques for multiplication modulo triangular sets

We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li et al. (2007), we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given notably to additions of algebraic numbers in small characteristic.

[1]  Marc Moreno Maza,et al.  Lifting techniques for triangular decompositions , 2005, ISSAC.

[2]  Marc Moreno Maza,et al.  High-Performance Symbolic Computation in a Hybrid Compiled-Interpreted Programming Environment , 2008, 2008 International Conference on Computational Sciences and Its Applications.

[3]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[4]  Daniel Reischert.  Asymptotically fast computation of subresultants , 1997, ISSAC.

[5]  Annick Valibouze,et al.  Computation of the Decomposition Group of a Triangular Ideal , 2004, Applicable Algebra in Engineering, Communication and Computing.

[6]  Hervé Perdry,et al.  Using Newton Identities in small characteristic: the Galois fields case , 2004 .

[7]  Éric Schost,et al.  A simple and fast algorithm for computing exponentials of power series , 2009, Inf. Process. Lett..

[8]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[9]  Joris van der Hoeven.  Newton's method and FFT trading , 2010, J. Symb. Comput..

[10]  Richard P. Brent,et al.  Multiple-precision zero-finding methods and the complexity of elementary function evaluation , 1975, ArXiv.

[11]  Arnold Schönhage,et al.  The fundamental theorem of algebra in terms of computational complexity - preliminary report , 1982 .

[12]  Annick Valibouze,et al.  Calcul de résolvantes avec les modules de Cauchy , 1999, Exp. Math..

[13]  Éric Schost,et al.  Construction of Secure Random Curves of Genus 2 over Prime Fields , 2004, EUROCRYPT.

[14]  Roberto Dvornicich,et al.  Newton Symmetric Functions and the Arithmetic of Algebraically Closed Fields , 1987, AAECC.

[15]  Lars Langemyr.  Algorithms for a Multiple Algebraic Extension II , 1991 .

[16]  Annick Valibouze,et al.  Using Galois Ideals for Computing Relative Resolvents , 1998, J. Symb. Comput..

[17]  Kazuhiro Yokoyama,et al.  A Modular Method for Computing the Splitting Field of a Polynomial , 2006, ANTS.

[18]  Martin Fürer.  Faster integer multiplication , 2007, STOC '07.

[19]  Mark van Hoeij,et al.  A modular GCD algorithm over number fields presented with multiple extensions , 2002, ISSAC '02.

[20]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[21]  Lars Langemyr.  Algorithms for a Multiple Algebraic Extension II , 1991, AAECC.

[22]  Victor Y. Pan.  Simple Multivariate Polynomial Multiplication , 1994, J. Symb. Comput..

[23]  Éric Schost,et al.  Evaluation Properties of Symmetric Polynomials , 2006, Int. J. Algebra Comput..

[24]  Marc Moreno Maza,et al.  On the complexity of the D5 principle , 2005, SIGS.

[25]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

[26]  Marc Moreno Maza,et al.  On computer-assisted classification of coupled integrable equations , 2001, ISSAC '01.

[27]  Éric Schost,et al.  Multivariate power series multiplication , 2005, ISSAC.

[28]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[29]  Joris van der Hoeven.  The truncated fourier transform and applications , 2004, ISSAC '04.

[30]  Joseph F. Traub.  Analytic Computational Complexity , 1976 .

[31]  Marc Moreno Maza,et al.  Computation of canonical forms for ternary cubics , 2002, ISSAC '02.

[32]  Marc Moreno Maza,et al.  Fast arithmetic for triangular sets: from theory to practice , 2007, ISSAC '07.

[33]  Kazuhiro Yokoyama,et al.  Multi-modular algorithm for computing the splitting field of a polynomial , 2008, ISSAC '08.

[34]  Alfred V. Aho,et al.  Evaluating Polynomials at Fixed Sets of Points , 1975, SIAM J. Comput..

[35]  Bernd Sturmfels,et al.  Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) , 2008 .

[36]  Philippe Flajolet,et al.  Fast computation of special resultants , 2006, J. Symb. Comput..

[37]  Armin Straub,et al.  The $p$-adic valuation of $k$-central binomial coefficients , 2009 .