Abstract
In this paper, we give a thorough revision of Lakshman’s paper by fixing some serious flaws in his approach. Furthermore, following this analysis, an intrinsic complexity bound for the construction of zero-dimensional Gröbner bases is given. Our complexity bound is in terms of the degree of the input ideal as well as the degrees of its generators. Finally, as an application of the presented method, we exhibit and analyze a (Monte Carlo) probabilistic algorithm to compute the degree of an equidimensional ideal.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
That is all isolated primes of \({\mathfrak {a}}\) share the same dimension. Note that, in general, an equidimensional is not unmixed. A proper ideal is said to be unmixed if its dimension is equal to the dimension of every associated prime of the ideal.
References
Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. In cooperation with Heinz Kredel. Springer, New York (1993). https://doi.org/10.1007/978-1-4612-0913-3
Belabas, K., van Hoeij, M., Klüners, J., Steel, A.: Factoring polynomials over global fields. J. Théor. Nombres Bordx. 21(1), 15–39 (2009)
Bost, J.B., Gillet, H., Soulé, C.: Un analogue arithmétique du théorème de Bézout. C. R. Acad. Sci. Paris Sér. I 312(11), 845–848 (1991)
Brownawell, W.D., Masser, D.W.: Multiplicity estimates for analytic functions. II. Duke Math. J. 47, 273–295 (1980)
Buchberger, B.: Bruno Buchberger’s PhD thesis 1965: an algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Translation from the German. J. Symb. Comput. 41(3–4), 475–511 (2006). https://doi.org/10.1016/j.jsc.2005.09.007
Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 4th edn. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16721-3
Dickenstein, A., Fitchas, N., Giusti, M., Sessa, C.: The membership problem for unmixed polynomial ideals is solvable in single exponential time. Discrete Appl. Math. 33(1–3), 73–94 (1991)
Durvye, C.: Algorithmes pour la décomposition primaire des idéaux polynomiaux de dimension nulle donnés en évaluation. Ph.D. thesis, University of Versailles-St Quentin en Yvelines (2008)
Durvye, C.: Evaluation techniques for zero-dimensional primary decomposition. J. Symb. Comput. 44(9), 1089–1113 (2009)
Durvye, C., Lecerf, G.: A concise proof of the Kronecker polynomial system solver from scratch. Expo. Math. 26(2), 101–139 (2008)
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)
Fernández, M., Pardo, L.M.: An arithmetic Poisson formula for the multi-variate resultant. J. Complexity 29(5), 323–350 (2013)
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). https://doi.org/10.1007/3-540-51082-6_83
Giusti, M., Lecerf, G., Salvy, B.: A Gröbner free alternative for polynomial system solving. J. Complexity 17(1), 154–211 (2001)
Hashemi, A., Heintz, J., Pardo, L.M., Solernó, P.: On Bézout inequalities for non-homogeneous polynomial ideals. arXiv:1701.04341 (2017)
Heintz, J., Schnorr, C.P.: Testing polynomials which are easy to compute. In: International Symposium on Logic and Algorithmic, Zürich 1980, Monographs of L’Enseignement Mathematique, vol. 30, pp. 237–254 (1982)
Heintz, J.: Definability and fast quantifier elimination in algebraically closed fields. Theor. Comput. Sci. 24, 239–277 (1983)
Kemper, G.: A Course in Commutative Algebra, vol. 256. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-03545-6
Lakshman, Y.N.: A single exponential bound on the complexity of computing Gröbner bases of zero dimensional ideals. In: Mora, T., Traverso, C. (eds.) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol. 94, pp. 227–234. Birkhäuser, Boston (1991). https://doi.org/10.1007/978-1-4612-0441-1_15
Lakshman, Y.N., Lazard, D.: On the complexity of zero-dimensional algebraic systems. In: Mora, T., Traverso, C. (eds.) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol. 94, pp. 217–225. Birkhäuser, Boston (1991)
Lazard, D.: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations. In: van Hulzen, J.A. (ed.) EUROCAL 1983. LNCS, vol. 162, pp. 146–156. Springer, Heidelberg (1983). https://doi.org/10.1007/3-540-12868-9_99
Lazard, D.: Résolution des systèmes d’équations algébriques. Theor. Comput. Sci. 15, 77–110 (1981). https://doi.org/10.1016/0304-3975(81)90064-5
Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC 2014, pp. 296–303. Association for Computing Machinery (ACM), New York (2014)
Lecerf, G.: Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complexity 19(4), 564–596 (2003)
Lelong, P.: Mesure de Mahler et calcul de constantes universelles pour les polynômes de \(n\) variables. Math. Ann. 299(4), 673–695 (1994)
Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46, 305–329 (1982). https://doi.org/10.1016/0001-8708(82)90048-2
McKinnon, D.: An arithmetic analogue of Bezout’s theorem. Compos. Math. 126(2), 147–155 (2001)
Pardo, L.M., Pardo, M.: On the zeta Mahler measure function of the Jacobian determinant, condition numbers and the height of the generic discriminant. Appl. Algebra Eng. Commun. Comput. 27(4), 303–358 (2016). https://doi.org/10.1007/s00200-016-0284-9
Philippon, P.: Critères pour l’indépendance algébrique. Publ. Math. Inst. Hautes Étud. Sci. 64, 5–52 (1986)
Philippon, P.: Sur des hauteurs alternatives. I. (On alternative heights. I). Math. Ann. 289(2), 255–283 (1991)
Philippon, P.: Sur des hauteurs alternatives. II. (On alternative heights. II). Ann. Inst. Fourier 44(4), 1043–1065 (1994)
Philippon, P.: Sur des hauteurs alternatives. III. J. Math. Pures Appl. (9) 74(4), 345–365 (1995)
Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach. 27, 701–717 (1980)
Hoeven, J.: On the complexity of multivariate polynomial division. In: Kotsireas, I.S., Martínez-Moro, E. (eds.) ACA 2015. SPMS, vol. 198, pp. 447–458. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56932-1_28
Van der Waerden, B.L.: Algebra. Volume II. Based in part on lectures by. E. Artin and E. Noether. Transl. from the German 5th ed. by John R. Schulenberger. Springer, New York (1991)
Vogel, W.: Lectures on results on Bezout’s theorem. Notes by D. P. Patil. Lectures on Mathematics and Physics. Mathematics, 74. Tata Institute of Fundamental Research. Springer, Berlin, ix, 132 p. (1984)
von zur Gathen, J., Panario, D.: Factoring polynomials over finite fields: a survey. J. Symb. Comput. 31(1–2), 3–17 (2001)
Zariski, O., Samuel, P.: Commutative algebra. Vol. II. The University Series in Higher Mathematics. Princeton, N.J.-Toronto-London-New York: D. Van Nostrand Company, Inc. x, 414 p. (1960)
Zippel, R.: Interpolating polynomials from their values. J. Symb. Comput. 9(3), 375–403 (1990)
Acknowledgments
The research was partially supported by the following Iranian, Argentinian and Spanish Grants:
– IPM Grant No. 98550413 (Amir Hashemi)
– UBACyT 20020170100309BA and PICT-2014-3260 (Joos Heintz)
– Spanish Grant MTM2014-55262-P (Luis M. Pardo)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Hashemi, A., Heintz, J., Pardo, L.M., Solernó, P. (2020). Intrinsic Complexity for Constructing Zero-Dimensional Gröbner Bases. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-60026-6_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-60025-9
Online ISBN: 978-3-030-60026-6
eBook Packages: Computer ScienceComputer Science (R0)