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Fast Gröbner basis computation and polynomial reduction for generic bivariate ideals

  • Joris van der HoevenEmail author
  • Robin Larrieu
Original Paper
  • 42 Downloads

Abstract

Let \(A, B \in \mathbb {K} [X, Y]\) be two bivariate polynomials over an effective field \(\mathbb {K}\), and let G be the reduced Gröbner basis of the ideal \(I :=\langle A, B \rangle \) generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, we design a quasi-optimal algorithm for the reduction of \(P \in \mathbb {K} [X, Y]\) modulo G, where “quasi-optimal” is meant in terms of the size of the input ABP. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra \(\mathbb {A} :=\mathbb {K} [X, Y] / \langle A, B \rangle \), both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.

Keywords

Polynomial reduction Gröbner basis Complexity Algorithm 

Mathematics Subject Classification

13P10 

Notes

Acknowledgements

We thank Vincent Neiger for a remark that simplified Algorithm 6. We also thank the anonymous referees for helpful comments and suggestions.

References

  1. 1.
    Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of the F5 Gröbner basis algorithm. J. Symb. Comput. 70, 1–24 (2014)zbMATHGoogle Scholar
  2. 2.
    Becker, T., Weispfenning, V.: Gröbner bases: a computational approach to commutative algebra. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Mathematics, vol. 141. Springer, New York (1993)CrossRefGoogle Scholar
  3. 3.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Ph.D. Thesis, Universitat Innsbruck, Austria (1965)Google Scholar
  4. 4.
    Cantor, D.G., Kaltofen, E.: On fast multiplication of polynomials over arbitrary algebras. Acta Inf. 28(7), 693–701 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC’02, pp. 75–83. ACM, New York (2002)Google Scholar
  7. 7.
    Faugère, J.-C., Gaudry, P., Huot, L., Renault, G.: Polynomial systems solving by fast linear algebra. arXiv:1304.6039 (2013)
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Fischer, M.J., Stockmeyer, L.J.: Fast on-line integer multiplication. In: Proceedings of the 5th ACM Symposium on Theory of Computing, vol. 9, pp. 67–72 (1974)Google Scholar
  10. 10.
    Fröberg, R., Hollman, J.: Hilbert series for ideals generated by generic forms. J. Symb. Comput. 17(2), 149–157 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Galligo, A.: A propos du théoreme de préparation de Weierstrass. In: Norguet, F. (ed.) Fonctions de plusieurs Variables Complexes, pp. 543–579. Springer, Berlin (1974)Google Scholar
  12. 12.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, Cambridge (2013)Google Scholar
  13. 13.
    Gerdt, V.P., Blinkov, Y.A.: Involutive bases of polynomial ideals. Math. Comput. Simul. 45(5), 519–541 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Harvey, D., van der Hoeven, J.: Faster polynomial multiplication over finite fields using cyclotomic coefficient rings. J. Complex (in press). https://doi.org/10.1016/j.jco.2019.03.004 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Harvey, D., van der Hoeven, J., Lecerf, G.: Faster polynomial multiplication over finite fields. J. ACM 63(6), 52 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    van der Hoeven, J.: Relax, but don’t be too lazy. JSC 34, 479–542 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    van der Hoeven, J.: Faster relaxed multiplication. In: Proceeding of the ISSAC’14, pp. 405–412, Kobe (July 2014)Google Scholar
  18. 18.
    van der Hoeven, J.: On the complexity of polynomial reduction. In: Kotsireas, I., Martínez-Moro, E. (eds.) Proceeding of the Applications of Computer Algebra 2015, Volume 198 of Springer Proceedings in Mathematics and Statistics, pp. 447–458. Springer, Cham (2015)Google Scholar
  19. 19.
    van der Hoeven, J., Larrieu, R.: Fast reduction of bivariate polynomials with respect to sufficiently regular Gröbner bases. In: Kauers, M., Ovchinnikov, A., Schost, É. (eds.) Proceeding of the ISSAC’18, pp. 199–206. ACM, New York (2018)Google Scholar
  20. 20.
    van der Hoeven, J., Schost, É.: Multi-point evaluation in higher dimensions. AAECC 24(1), 37–52 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lebreton, R., Schost, E., Mehrabi, E.: On the complexity of solving bivariate systems: the case of non-singular solutions. In: ISSAC: International Symposium on Symbolic and Algebraic Computation, pp. 251–258, Boston (June 2013)Google Scholar
  22. 22.
    Mayr, E.: Membership in polynomial ideals over \(Q\) is exponential space complete. STACS 89, 400–406 (1989)MathSciNetGoogle Scholar
  23. 23.
    Moreno-Socías, G.: Degrevlex Gröbner bases of generic complete intersections. J. Pure Appl. Algebra 180(3), 263–283 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schönhage, A.: Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Inf. 7, 395–398 (1977)CrossRefGoogle Scholar
  25. 25.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)MathSciNetCrossRefGoogle Scholar
  26. 26.
    The Sage Developers: SageMath, the Sage Mathematics Software System (Version 8.0) (2017). https://www.sagemath.org

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire d’informatique de l’École polytechnique, LIX, UMR 7161CNRS & École polytechniquePalaiseauFrance

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