research-article

Algorithms for the universal decomposition algebra

Authors:

Romain Lebreton,

Éric SchostAuthors Info & Claims

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Pages 234 - 241

https://doi.org/10.1145/2442829.2442864

Published: 22 July 2012 Publication History

Abstract

Let k be a field and let f ∈ k [T] be a polynomial of degree n. The universal decomposition algebra A is the quotient of k [X₁,...,X_n] by the ideal of symmetric relations (those polynomials that vanish on all permutations of the roots of f). We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an isomorphism of the form A k[T]/Q(T), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A.

References

[1]

A. V. Aho, K. Steiglitz, and J. D. Ullman. Evaluating polynomials at fixed sets of points. SIAM J. Comput., 4:533--539, 1975.

Digital Library

[2]

J.-M. Arnaudiès and A. Valibouze. Calculs de résolvantes. Rapport LITP 94.46, 1994.

[3]

J.-M. Arnaudiès and A. Valibouze. Lagrange resolvents. J. P. Appl. Alg., 117/118:23--40, 1997.

[4]

P. Aubry and A. Valibouze. Using Galois ideals for computing relative resolvents. J. Symb. Comp., 30(6):635--651, 2000.

Digital Library

[5]

P. Aubry and A. Valibouze. Algebraic computation of resolvents without extraneous powers. European Journal of Combinatorics, 2012. To appear.

Digital Library

[6]

A. Bostan. Algorithmique efficace pour des opérations de base en Calcul formel. PhD thesis, École Polytechnique, 2003.

[7]

A. Bostan, M. F. I. Chowdhury, J. van der Hoeven, and É. Schost. Homotopy methods for multiplication modulo triangular sets. J. Symb. Comp., 2011. To appear.

Digital Library

[8]

A. Bostan, P. Flajolet, B. Salvy, and É. Schost. Fast computation of special resultants. J. Symb. Comp., 41:1--29, 2006.

Digital Library

[9]

A. Bostan and É. Schost. Polynomial evaluation and interpolation on special sets of points. J. Complex., 21:420--446, 2005.

Digital Library

[10]

F. Boulier, F. Lemaire, and M. Moreno Maza. Pardi! In ISSAC'01, pp. 38--47. ACM, 2001.

Digital Library

[11]

N. Bourbaki. Éléments de mathématique, Fasc. XXIII. Hermann, Paris, 1973. Livre II: Algèbre. Chapitre 8.

[12]

D. G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Inform., 28:693--701, 1991.

Digital Library

[13]

D. Casperson and J. McKay. Symmetric functions, m-sets, and Galois groups. Math. Comp., 63(208):749--757, 1994.

Digital Library

[14]

N. Chebotarev. Grundzüge des Galois'shen Theorie. P. Noordhoff, 1950.

[15]

A. Colin and M. Giusti. Efficient computation of squarefree Lagrange resolvents. 2010.

[16]

D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comp., 9(3):251--280, 1990.

Digital Library

[17]

X. Dahan, M. Moreno Maza, É. Schost, and Y. Xie. On the complexity of the D5 principle. In TC'06, pp. 149--168, 2006.

[18]

L. De Feo and É. Schost. Fast arithmetics in Artin-Schreier towers over finite fields. In ISSAC'09, pp. 127--134. ACM, 2009.

Digital Library

[19]

C. Durvye and G. Lecerf. A concise proof of the Kronecker polynomial system solver from scratch. Expo. Math., 26:101--139, 2008.

[20]

J. C. Faugère, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symb. Comp., 16:329--344, 1993.

Digital Library

[21]

J.-C. Faugère and C. Mou. Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices. In ISSAC'11, pp. 115--122. ACM, 2011.

Digital Library

[22]

J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, Cambridge, second edition, 2003.

Digital Library

[23]

M. Giusti, G. Lecerf, and B. Salvy. A Gröbner-free alternative for polynomial system solving. J. Complex., 17:154--211, 2001.

Digital Library

[24]

J. Heintz, T. Krick, S. Puddu, J. Sabia, and A. Waissbein. Deformation techniques for efficient polynomial equation solving. J. Complex., 16(1):70--109, 2000.

Digital Library

[25]

J. König. Aus dem Ungarischen übertragen vom Verfasser. B. G. Teubner, Leipzig, 1903.

[26]

L. Kronecker. Grundzüge einer arithmetischen theorie des algebraischen grössen. J. reine angew. Math., 92:1--122, 1882.

[27]

J.-L. Lagrange. Réflexions sur la résolution algébrique des équations. Mémoires de l'Académie de Berlin, 1770.

[28]

F. Lehobey. Resolvent computations by resultants without extraneous powers. In ISSAC'97, pp. 85--92. ACM, 1997.

Digital Library

[29]

X. Li, M. Moreno Maza, and W. Pan. Computations modulo regular chains. In ISSAC'09, pp. 239--246. ACM, 2009.

Digital Library

[30]

X. Li, M. Moreno Maza, and É. Schost. Fast arithmetic for triangular sets: from theory to practice. J. Symb. Comp., 44(7):891--907, 2009.

Digital Library

[31]

F. S. Macaulay. The algebraic theory of modular systems. Cambridge University Press, 1994.

[32]

A. Poteaux and É. Schost. On the complexity of computing with zero-dimensional triangular sets. Submitted, 2011.

[33]

N. Rennert. A parallel multi-modular algorithm for computing Lagrange resolvents. J. Symb. Comput., 37(5):547--556, 2004.

[34]

N. Rennert and A. Valibouze. Calcul de résolvantes avec les modules de Cauchy. Exp. Math., 8(4):351--366, 1999.

[35]

F. Rouillier. Solving zero-dimensional systems through the rational univariate representation. Appl. Alg. Engrg. Comm. Comput., 9(5):433--461, 1999.

[36]

A. Schönhage. The fundamental theorem of algebra in terms of computational complexity. Technical report, U. Tübingen, 1982.

[37]

A. Schönhage and V. Strassen. Schnelle Multiplikation grosser Zahlen. Computing, 7:281--292, 1971.

[38]

J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701--717, 1980.

Digital Library

[39]

V. Shoup. Fast construction of irreducible polynomials over finite fields. J. Symb. Comp., 17(5):371--391, 1994.

Digital Library

[40]

L. Soicher. The computation of the Galois groups. PhD thesis, Concordia University, Montreal, Quebec, Canada, 1981.

[41]

A. Stothers. On the Complexity of Matrix Multiplication. PhD thesis, University of Edinburgh, 2010.

[42]

A. Valibouze. Fonctions symétriques et changements de bases. In EUROCAL'87, vol. 378 of LNCS, pp. 323--332, 1989.

Digital Library

[43]

V. Vassilevska Williams. Breaking the Coppersmith-Winograd barrier. 2011.

[44]

K. Yokoyama. A modular method for computing the Galois groups of polynomials. J. P. Appl. Alg., 117/118:617--636, 1997.

[45]

R. Zippel. Probabilistic algorithms for sparse polynomials. In EUROSAM'79, vol. 72 of LNCS, pp. 216--226. Springer, 1979.

Digital Library

Cited By

Moroz GSchost EAbramov SZima EGao X(2016)A Fast Algorithm for Computing the Truncated ResultantProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930931(341-348)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930931
De Feo LDoliskani JSchost EMonagan MCooperman GGiesbrecht M(2013)Fast algorithms for l-adic towers over finite fieldsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465956(165-172)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465956

Index Terms

Algorithms for the universal decomposition algebra
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Representation of mathematical objects
    2. Symbolic and algebraic algorithms

Recommendations

Two-Dimensional Universal Algebra / Dvoudimensionální Univerzální Algebra
Polynomial Decomposition Algorithms
Applying Universal Algebra to Lambda Calculus*

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

July 2012

390 pages

ISBN:9781450312691

DOI:10.1145/2442829

General Chair:
Joris van der Hoeven
École Polytechnique, France
,
Program Chair:
Mark van Hoeij
Florida State U.

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Grenoble University: Grenoble University
INRIA: Institut Natl de Recherche en Info et en Automatique

In-Cooperation

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 July 2012

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Research-article

Conference

ISSAC'12

Sponsor:

Grenoble University
INRIA

ISSAC'12: International Symposium on Symbolic and Algebraic Computation

July 22 - 25, 2012

Grenoble, France

Acceptance Rates

ISSAC '12 Paper Acceptance Rate 46 of 86 submissions, 53%;

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
51
Total Downloads

Downloads (Last 12 months)1
Downloads (Last 6 weeks)0

Reflects downloads up to 06 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Moroz GSchost EAbramov SZima EGao X(2016)A Fast Algorithm for Computing the Truncated ResultantProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930931(341-348)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930931
De Feo LDoliskani JSchost EMonagan MCooperman GGiesbrecht M(2013)Fast algorithms for l-adic towers over finite fieldsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465956(165-172)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465956

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents