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Algorithms for Zero-Dimensional Ideals Using Linear Recurrent Sequences

  • Vincent Neiger
  • Hamid Rahkooy
  • Éric Schost
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10490)

Abstract

Inspired by Faugère and Mou’s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing of the annihilator of one or several such sequences.

Notes

Acknowledgements

We thank the reviewers for their remarks and suggestions. The third author is supported by an NSERC Discovery Grant.

References

  1. 1.
    Berthomieu, J., Boyer, B., Faugère, J.-C.: Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences. J. Symb. Comput. 83, 36–67 (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berthomieu, J., Faugère, J.-C.: Guessing linear recurrence relations of sequence tuples and P-recursive sequences with linear algebra. In: ISSAC 2016, pp. 95–102. ACM (2016)Google Scholar
  3. 3.
    Berthomieu, J., Faugère, J.-C.: In-depth comparison of the Berlekamp-Massey-Sakata and the Scalar-FGLM algorithms: the non adaptive variants. hal-01516708, May 2017Google Scholar
  4. 4.
    Bostan, A., Salvy, B., Schost, É.: Fast algorithms for zero-dimensional polynomial systems using duality. AAECC 14, 239–272 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11), 1851–1872 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dahan, X., Moreno Maza, M., Schost, É., Xie, Y.: On the complexity of the D5 principle. In: Transgressive Computing, pp. 149-168 (2006)Google Scholar
  7. 7.
    Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 289–290. Springer, Heidelberg (1985). doi: 10.1007/3-540-15984-3_279 CrossRefGoogle Scholar
  8. 8.
    Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry, vol. 150. Springer Science & Business Media, New York (2013). doi: 10.1007/978-1-4612-5350-1 zbMATHGoogle Scholar
  9. 9.
    Faugère, J.-C., Gaudry, P., Huot, L., Renault, G.: Polynomial Systems Solving by Fast Linear Algebra (2013). https://hal.archives-ouvertes.fr/hal-00816724
  10. 10.
    Faugère, J.-C., Gaudry, P., Huot, L., Renault, G.: Sub-cubic change of ordering for Gröbner basis: a probabilistic approach. In: ISSAC 2014, pp. 170-177. ACM (2014)Google Scholar
  11. 11.
    Faugère, J.-C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symb. Comput. 16(4), 329–344 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Faugère, J.-C., Mou, C.: Sparse FGLM algorithms. J. Symb. Comput. 80(3), 538–569 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gianni, P., Mora, T.: Algebrric solution of systems of polynomirl equations using Groebher bases. In: Huguet, L., Poli, A. (eds.) AAECC 1987. LNCS, vol. 356, pp. 247–257. Springer, Heidelberg (1989). doi: 10.1007/3-540-51082-6_83 CrossRefGoogle Scholar
  15. 15.
    Gröbner, W.: Über irreduzible Ideale in kommutativen Ringen. Math. Ann. 110(1), 197–222 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Macaulay, F.S.: Modern algebra and polynomial ideals. Math. Proc. Camb. Philos. Soc. 30(1), 27–46 (1934)CrossRefzbMATHGoogle Scholar
  17. 17.
    Marinari, M.G., Mora, T., Möller, H.M.: Gröbner bases of ideals defined by functionals with an application to ideals of projective points. AAECC 4, 103–145 (1993)CrossRefzbMATHGoogle Scholar
  18. 18.
    Moreno-Socías, G.: Autour de la fonction de Hilbert-Samuel (escaliers d’ideaux polynomiaux). Ph.D. thesis, École polytechnique (1991)Google Scholar
  19. 19.
    Mourrain, B.: Fast algorithm for border bases of Artinian Gorenstein algebras. ArXiv e-prints, May 2017Google Scholar
  20. 20.
    Neiger, V.: Bases of relations in one or several variables: fast algorithms and applications. Ph.D. thesis, École Normale Supérieure de Lyon, November 2016Google Scholar
  21. 21.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. AAECC 9(5), 433–461 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sakata, S.: Extension of the Berlekamp-Massey algorithm to \(N\) dimensions. Inform. Comput. 84(2), 207–239 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shoup, V.: A new polynomial factorization algorithm and its implementation. J. Symb. Comput. 20(4), 363–397 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark
  2. 2.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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