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Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality

  • Alin Bostan
  • Bruno Salvy
  • Éric Schost
Article

Abstract

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X 1 ,…,X n ]/ℐ, where ℐ is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space \(\widehat{A}\). An important feature of our algorithms is that we do not require \(\widehat{A}\) to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 nD 5/2 ) and O(n2 nD 5/2 ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.

Keywords

Duality Polynomial system solving Linear recurrent sequences 

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References

  1. 1.
    Alonso, M.-E., Becker, E., Roy, M.-F., Wörmann, T.: Zeros, multiplicities, and idempotents for zero-dimensional systems. In: Algorithms in algebraic geometry and applications (Santander, 1994), Birkhäuser, Basel, 1996, pp. 1–15Google Scholar
  2. 2.
    Antoniou, A.: Digital Filters: Analysis and Design. McGraw-Hill Book Co., 1979Google Scholar
  3. 3.
    Arnaudiès, J.-M., Valibouze, A.: Lagrange resolvents. J. Pure Appl. Algebra 117/118, 23–40, 1997, Algorithms for algebra (Eindhoven, 1996)Google Scholar
  4. 4.
    Becker, E., Cardinal, J. P., Roy, M.-F., Szafraniec, Z.: Multivariate Bezoutians, Kronecker symbol and Eisenbud-Levine formula. In: Algorithms in algebraic geometry and applications (Santander, 1994), Birkhäuser, Basel, 1996, pp. 79–104Google Scholar
  5. 5.
    Becker, E., Wörmann, T.: Radical computations of zero-dimensional ideals and real root counting. Mathematics and Computers in Simulation, 42(4-6), 561–569 (1996), Symbolic computation, new trends and developments (Lille, 1993)Google Scholar
  6. 6.
    Berlekamp, E. R.: Algebraic coding theory. McGraw-Hill Book Co., New York, 1968Google Scholar
  7. 7.
    Björck, G.: Functions of modulus 1 on Z n whose Fourier transforms have constant modulus, and ‘‘cyclic n-roots’’. In: Recent advances in Fourier analysis and its applications (Il Ciocco, 1989), Volume 315 of NATO Advance Science Institutes Series C: Mathematical and Physical Sciences, Kluwer Academic Publishers, Dordrecht, 1990, pp. 131–140Google Scholar
  8. 8.
    Bommer, R.: High order derivations and primary ideals to regular prime ideals. Arch. Math. 46(6), 511–521 (1986)zbMATHGoogle Scholar
  9. 9.
    Bordewijk, J. L.: Inter-reciprocity applied to electrical networks. Appl. Sci. Res. B 1, 1–74 (1956)zbMATHGoogle Scholar
  10. 10.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. Journal of Symbolic Computation 24(3-4), 235–265 (1997) See also http://www.maths.usyd.edu.au:8000/u/magma/Google Scholar
  11. 11.
    Bostan, A., Lecerf, G., Schost, É.: Tellegen’s principle into practice. In: Symbolic and Algebraic Computation, 2003, Proceedings ISSAC’03, Philadelphia, August 2003. To appearGoogle Scholar
  12. 12.
    Brent, R. P., Kung, H. T.: Fast algorithms for manipulating formal power series. J. ACM 25(4), 581–595, October 1978CrossRefzbMATHGoogle Scholar
  13. 13.
    Buchberger, B.: Gröbner bases: An algorithmic method in polynomial ideal theory. In: Multidimensional System Theory, Reidel, Dordrecht, 1985, pp. 374–383Google Scholar
  14. 14.
    Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory. Springer, 1997Google Scholar
  15. 15.
    Charlap, L. S., Coley, R., Robbins, D.: Enumeration of rational points on elliptic curves over finite fields. Preprint, 1991Google Scholar
  16. 16.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbolic Comput. 9(3), 251–280, March 1990zbMATHGoogle Scholar
  17. 17.
    Cox, D., Little, J., O’Shea, D.: Using algebraic geometry. Springer-Verlag, New York, 1998Google Scholar
  18. 18.
    Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. Graduate Texts in Mathematics. Springer-Verlag, New York, 1995Google Scholar
  19. 19.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). J. Pure Appl. Algebra, 139(1–3), 61–88 (1999) Proceedinds of MEGA’98Google Scholar
  20. 20.
    Faugère, J.-C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comput. 16(4), 329–344 (1993)CrossRefGoogle Scholar
  21. 21.
    Fiduccia, C. M.: On obtaining upper bounds on the complexity of matrix multiplication. In: Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, pp. 31–40, 187–212Google Scholar
  22. 22.
    Fiduccia, C. M.: On the algebraic complexity of matrix multiplication. PhD thesis, Brown Univ., Providence, RI, Center Comput. Inform. Sci., Div. Engin., 1973Google Scholar
  23. 23.
    Gaudry, P.: Algorithmique des courbes hyperelliptiques et applications à la cryptologie. PhD thesis, École polytechnique, 2000Google Scholar
  24. 24.
    Gaudry, P., Schost, É.: Modular equations for hyperelliptic curves. Technical report, École polytechnique, 2002Google Scholar
  25. 25.
    Giusti, M., Heintz, J., Hägele, K., Morais, J. E., Pardo, L. M., Montaña, J. L.: Lower bounds for Diophantine approximations. J. Pure Appl. Algebra 117/118, 277–317 (1997), Algorithms for algebra (Eindhoven, 1996)Google Scholar
  26. 26.
    Giusti, M., Heintz, J., Morais, J. E., Morgenstern, J., Pardo, L. M.: Straight-line programs in geometric elimination theory. J. Pure Appl. Algebra 124, 101–146 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Giusti, M., Lecerf, G., Salvy, B.: A Gröbner free alternative for polynomial system solving. J. Complexity 17(1), 154–211 (2001)CrossRefzbMATHGoogle Scholar
  28. 28.
    Göttfert, R., Niederreiter., H.: On the minimal polynomial of the product of linear recurring sequences. Finite Fields and their Applications 1(2), 204–218 (1995), Special issue dedicated to Leonard CarlitzCrossRefGoogle Scholar
  29. 29.
    Gröbner, W.: La théorie des idéaux et la géométrie algébrique. In: Deuxième Colloque de Géométrie Algébrique, Liège, 1952, Georges Thone, Liège, 1952, pp. 129–144Google Scholar
  30. 30.
    Hopcroft, J., Musinski, J.: Duality applied to the complexity of matrix multiplication and other bilinear forms. SIAM J. Comput. 2, 1973Google Scholar
  31. 31.
    Kaltofen, E.: Analysis of Coppersmith’s block Wiedemann algorithm for the parallel solution of sparse linear systems. Math. Comput. 64(210), 777–806 (1995)zbMATHGoogle Scholar
  32. 32.
    Kaltofen, E., Corless, R. M., Jeffrey, D.J.: Challenges of symbolic computation: my favorite open problems. J. Symbolic Comput. 29(6), 891–919 (2000)CrossRefzbMATHGoogle Scholar
  33. 33.
    Kaminski, M., Kirkpatrick, D. G., Bshouty, N. H.: Addition requirements for matrix and transposed matrix products. J. Algorithms 9(3), 354–364 (1988)zbMATHGoogle Scholar
  34. 34.
    Kreuzer, M., Robbiano, L.: Computational commutative algebra. 1. Springer-Verlag, Berlin, 2000Google Scholar
  35. 35.
    Kronecker, L.: Grundzüge einer arithmetischen Theorie der algebraischen Grössen. Journal für die reine und angewandte Mathematik 92, 1–122 (1882)Google Scholar
  36. 36.
    Kunz, E.: Kähler differentials. Vieweg advanced lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1986Google Scholar
  37. 37.
    Kurosh, A.: Cours d’algèbre supérieure. Éditions Mir, Moscou, 1973Google Scholar
  38. 38.
    Lang, S.: Introduction to algebraic geometry. Interscience Publishers, New York, 1958Google Scholar
  39. 39.
    Lang, S.: Algebra, Volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002Google Scholar
  40. 40.
    Lazard, D.: Solving zero-dimensional algebraic systems. J. Symbolic Comput. 13, 117–133 (1992)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Lazard, D., Valibouze, A.: Computing subfields: reverse of the primitive element problem. In: Computational algebraic geometry (Nice, 1992), Birkhäuser Boston, Boston, MA, 1993, pp. 163–176Google Scholar
  42. 42.
    Lecerf, G.: Une alternative aux méthodes de réécriture pour la résolution des systèmes algébriques. PhD thesis, École polytechnique, 2001Google Scholar
  43. 43.
    Lecerf, G.: Quadratic Newton iteration for systems with multiplicity. J. FoCM 2(3), 247–293 (2002)zbMATHGoogle Scholar
  44. 44.
    Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complexity, 2003. To appearGoogle Scholar
  45. 45.
    Little, J.: A key equation and the computation of error values for codes from order domains. Available at http://arXiv.org/math.AC/0303299, 2003Google Scholar
  46. 46.
    Macaulay, F. S.: The Algebraic Theory of Modular Systems. Cambridge University Press, 1916Google Scholar
  47. 47.
    Mallat, S.: Foveal detection and approximation for singularities. Applied and Computational Harmonic Analysis, 2003. To appearGoogle Scholar
  48. 48.
    Marinari, M. G., Mora, T., Möller, H. M.: Gröbner bases of ideals defined by functionals with an application to ideals of projective points. Applicable Algebra in Engineering, Communication and Computing 4, 103–145 (1993)Google Scholar
  49. 49.
    Massey, J. L.: Shift-register synthesis and BCH decoding. IEEE Transactions on Information Theory, IT-15, 122–127, 1969Google Scholar
  50. 50.
    Mourrain, B.: Isolated points, duality and residues. J. Pure Appl. Algebra 117/118, 469–493 (1997) Algorithms for algebra (Eindhoven, 1996)Google Scholar
  51. 51.
    Mourrain, B., Pan, V. Y.: Solving special polynomial systems by using structured matrices and algebraic residues. In: Foundations of computational mathematics (Rio de Janeiro, 1997), Springer, Berlin, 1997, pp. 287–304Google Scholar
  52. 52.
    Mourrain, B., Pan, V. Y.: Asymptotic acceleration of solving multivariate polynomial systems of equations. In: Proceedings STOC, ACM Press, 1998, pp. 488–496Google Scholar
  53. 53.
    Mourrain, B., Pan, V. Y.: Multivariate polynomials, duality, and structured matrices. J. Complexity 16(1), 110–180 (2000)CrossRefzbMATHGoogle Scholar
  54. 54.
    Mourrain, B., Pan, V. Y., Ruatta, O.: Accelerated solution of multivariate polynomial systems of equations. SIAM J. Comput. 32(2), 435–454 (2003)CrossRefGoogle Scholar
  55. 55.
    Nakai, Y.: High order derivations. I. Osaka J. Math. 7, 1–27 (1970)Google Scholar
  56. 56.
    Oberst, U.: The construction of Noetherian operators. J. Algebra 222(2), 595–620 (1999)CrossRefzbMATHGoogle Scholar
  57. 57.
    Osborn, H.: Modules of differentials. II. Math. Ann. 175, 146–158 (1968)zbMATHGoogle Scholar
  58. 58.
    Paterson, M. S., Stockmeyer, L. J.: On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. Comput. 2(1), 60–66, March 1973zbMATHGoogle Scholar
  59. 59.
    Penfield, Jr. P., Spencer, R., Duinker, S.: Tellegen’s theorem and electrical networks. The M.I.T. Press, Cambridge, Mass.-London, 1970Google Scholar
  60. 60.
    Rouillier, F.: Solving zero-dimensional systems through the Rational Univariate Representation. Applicable Algebra in Engineering, Communication and Computing 9(5), 433–461 (1999)Google Scholar
  61. 61.
    Saito, M., Sturmfels, B., Takayama, N.: Gröbner deformations of hypergeometric differential equations. Springer-Verlag, Berlin, 2000Google Scholar
  62. 62.
    Samuel, P.: Théorie algébrique des nombres. Hermann, 1971Google Scholar
  63. 63.
    Scheja, G., Storch, U.: Über Spurfunktionen bei vollständigen Durchschnitten. Journal für die reine und angewandte Mathematik 278/279, 174–190 (1975)Google Scholar
  64. 64.
    Schost, É.: Sur la résolution des systèmes polynomiaux à paramètres. PhD thesis, École polytechnique, 2000Google Scholar
  65. 65.
    Schwartz, J. T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717, October 1980CrossRefzbMATHGoogle Scholar
  66. 66.
    Shoup, V.: Fast construction of irreducible polynomials over finite fields. J. Symbolic Comput. 17(5), 371–391 (1994)CrossRefzbMATHGoogle Scholar
  67. 67.
    Shoup, V.: Efficient computation of minimal polynomials in algebraic extensions of finite fields. In: Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), New York, 1999, pp. 53–58 ACMGoogle Scholar
  68. 68.
    Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)zbMATHGoogle Scholar
  69. 69.
    Tellegen, B.: A general network theorem, with applications. Philips Research Reports 7, 259–269 (1952)zbMATHGoogle Scholar
  70. 70.
    Thiong~Ly., J.-A.: Note for computing the minimum polynomial of elements in large finite fields. In: Coding theory and applications (Toulon, 1988), Volume 388 of Lecture Notes in Comput. Sci., Springer, New York, 1989, pp. 185–192Google Scholar
  71. 71.
    von~zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, New York, 1999Google Scholar
  72. 72.
    Wiedemann, D.: Solving sparse linear equations over finite fields. IEEE Transactions on informations theory IT-32, 54–62, 1986Google Scholar
  73. 73.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Symbolic and algebraic computation, number~72 in Lecture Notes in Computer Science, Berlin, 1979. Springer. Proceedings EUROSAM ‘79, Marseille, 1979, pp. 216–226Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Laboratoire STIXÉcole polytechnique, IMARFrance, Romania
  2. 2.Projet AlgorithmesFrance
  3. 3.Laboratoire STIXÉcole polytechniqueFrance

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