Article

Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields

Authors:

Erich Kaltofen,

Pascal KoiranAuthors Info & Claims

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

Pages 162 - 168

https://doi.org/10.1145/1145768.1145798

Published: 09 July 2006 Publication History

Abstract

We present algorithms that compute all irreducible factors of degree ≤ d of supersparse (lacunary) multivariate polynomials in n variables over an algebraic number field in deterministic polynomial-time in (l+d)ⁿ, where l is the size of the input polynomial. In supersparse polynomials, the term degrees enter logarithmically as their numbers of binary digits into the size measure l. The factors are again represented as supersparse polynomials. If the factors are represented as straight-line programs or black box polynomials, we can achieve randomized polynomial-time in (l+d)^O(1). Our approach follows that by H. W. Lenstra, Jr., on computing factors of univariate supersparse polynomials over algebraic number fields. We generalize our ISSAC 2005 results for computing linear factors of supersparse bivariate polynomials over the rational numbers by appealing to recent lower bounds on the height of algebraic numbers and to a special case of the former Lang conjecture.

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Cited By

Roche DKauers MOvchinnikov ASchost É(2018)What Can (and Can't) we Do with Sparse Polynomials?Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209027(25-30)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209027
Grenet B(2016)LacunaryxACM Communications in Computer Algebra10.1145/2893803.289380749:4(121-124)Online publication date: 17-Feb-2016
https://dl.acm.org/doi/10.1145/2893803.2893807
Grenet B(2016)Bounded-degree factors of lacunary multivariate polynomialsJournal of Symbolic Computation10.1016/j.jsc.2015.11.01375:C(171-192)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.013
Show More Cited By

Index Terms

Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

July 2006

374 pages

ISBN:1595932763

DOI:10.1145/1145768

General Chair:
Barry Trager
IBM T.J. Watson Research Center, USA

Copyright © 2006 ACM.

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Published: 09 July 2006

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July 9 - 12, 2006

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Roche DKauers MOvchinnikov ASchost É(2018)What Can (and Can't) we Do with Sparse Polynomials?Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209027(25-30)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209027
Grenet B(2016)LacunaryxACM Communications in Computer Algebra10.1145/2893803.289380749:4(121-124)Online publication date: 17-Feb-2016
https://dl.acm.org/doi/10.1145/2893803.2893807
Grenet B(2016)Bounded-degree factors of lacunary multivariate polynomialsJournal of Symbolic Computation10.1016/j.jsc.2015.11.01375:C(171-192)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.013
Grenet BNagasaka KWinkler FSzanto A(2014)Computing low-degree factors of lacunary polynomialsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608649(224-231)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608649
Wu WChen JFeng Y(2014)Sparse bivariate polynomial factorizationScience China Mathematics10.1007/s11425-014-4850-y57:10(2123-2142)Online publication date: 26-Jun-2014
https://doi.org/10.1007/s11425-014-4850-y
Kaltofen E(2014)Symbolic Computation and Complexity Theory Transcript of My TalkComputer Mathematics10.1007/978-3-662-43799-5_1(3-7)Online publication date: 1-Oct-2014
https://doi.org/10.1007/978-3-662-43799-5_1
Mullen GPanario D(2013)CryptographyHandbook of Finite Fields10.1201/b15006-22(777-860)Online publication date: 17-Jun-2013
https://doi.org/10.1201/b15006-22
Chattopadhyay AGrenet BKoiran PPortier NStrozecki YMonagan MCooperman GGiesbrecht M(2013)Factoring bivariate lacunary polynomials without heightsProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465932(141-148)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1145/2465506.2465932
Kaltofen ENehring MSchost ÉEmiris I(2011)Supersparse black box rational function interpolationProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993916(177-186)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993916
Pébay PMaurice Rojas JThompson D(2011)Optimizing n-variate (n+k)-nomials for small kTheoretical Computer Science10.1016/j.tcs.2010.11.053412:16(1457-1469)Online publication date: 1-Apr-2011
https://dl.acm.org/doi/10.1016/j.tcs.2010.11.053
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