Abstract
LetF be a unimodularr×s matrix with entries beingn-variate polynomials over an infinite fieldK. Denote by deg(F) the maximum of the degrees of the entries ofF and letd=1+deg(F). We describe an algorithm which computes a unimodulars×s matrixM with deg(M)=(rd) O(n) such thatFM=[I r ,O], where [I r ,O] denotes ther×s matrix obtained by adding to ther×r unit matrixI r s−r zero columns.
We present the algorithm as an arithmetic network with inputs fromK, and we count field operations and comparisons as unit cost.
The sequential complexity of our algorithm amounts to\(S^{O(r^2 )} r^{O(n^2 )} d^{O(n^2 + r^2 )} \) field operations and comparisons inK whereas its parallel complexity isO(n 4 r 4log2(srd)).
The complexity bounds and the degree bound for deg(M) mentioned above are optimal in order. Our algorithm is inspired by Suslin's proof of Serre's Conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Bayer andM. Stillman, On the complexity of computing syzygies,J. Symbolic Comp. 6 (1988), 135–147.
S. Berkowitz, On computing the determinant in small parallel time using a small number of processors,Inform. Processing Letters 18 (1984), 147–150.
W. D. Brownawell, Bounds for the degree in the Nullstellensatz,Ann. Math. (Second Series)126 (3) (1987), 577–591.
B. Buchberger, Gröbner-Bases: an algorithmic method in polynomial ideal theory, inMultidimensional Systems Theory, N. K. Bose, ed., Reidel Publishing Company, Dordrecht, 1985, 184–232.
L. Caniglia, A. Galligo, and J. Heintz, Some new effectivity bounds in computational geometry, inProc. 6-th Int. Conf. Applied Algebra, Algebraic Algorithmic and Error Correcting Codes (AAECC-6), T. Mora, ed., Springer LN Comp. Sci.357 (1989), 131–151.
L. Caniglia, J. A. Guccione, and J. J. Guccione, Local Membership Problems for Polynomial Ideals, inEffective Methods in Algebraic Geometry, T. Mora and C. Traverso, eds., Progress in Math.94, Birkhäuser (1991), 31–45.
F. Chaqui,Algorithme de calcul d'une base pour les modules projectifs sur K[X,Y] et ℤ[X], Thèse 3ème Cycle, Université Paris-Sud, Centre d'Orsay (1983).
A. Chistov, Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic, inProc. Int. Conf. FCT 1985, Springer LN Comp. Sci.199 (1985), 63–69.
N. Fitchas andA. Galligo, Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel,Math. Nachr. 149 (1990), 231–253.
N. Fitchas, A. Galligo and J. Morgenstern, Algorithmes rapides en séquentiel et en parallèle pour l'élimination des quantificateurs en géométrie élémentaire, in F. Delon, M. Dickmann, D. Gondard (eds.),Séminaire Structures Algébriques Ordonnées: Sélection d'exposés 1986–1987. Publ. Université Paris VII32 (1990), 103–145.
F. R. Gantmacher,Teorya matrits. English version:The theory of matrices, Volume I, Chelsea Publishing Company, New York (1977).
J von zur Gathen, Parallel arithmetic computations: a survey,Proc. 13-th Symp. MFCS 1986, Springer LN Comp. Sci.233 (1986), 93–112.
G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale,Math. Ann. 95 (1926), 736–788.
J. Kollár, Sharp effective Nullstellensatz,J. Am. Math. Soc. 1 (1988), 963–975.
E. Kunz,Einführnug in die kommutative Algebra und algebraische Geometrie, F. Vieweg und Söhne, Braunschweig-Wiesbaden (1980).
T. Y. Lam,Serre's Conjecture, Springer LN Math.635 Springer-Verlag (1978).
A. Logar andB. Sturmfels, Algorithms for Quillen-Suslin Theorem,J. Algebra 145 (1992), 231–239.
E. Mayr andA. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals,Advances in Math. 46 (1982), 305–329.
K. Mulmuley, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field,Proc. 18-th Ann. ACM Symp. Theory of Comput. (1986), 338–339.
P. Philippon,Théorème des zéros effectif d'après J.Kollár, Probl. Dioph. 1988–89, Publ. Math. Univ. Paris VI88 (1988).
S. Seidenberg, Constructions in AlgebraTrans. Amer. Math. Soc. 197 (1974), 273–313.
J. P. Serre, Faisceaux algébriques cohérents,Amer. Math. 61 (1955), 191–274.
B. Teissier, Résultats récents d'algèbre commutative effective,Séminaire Bourbaki. Volume 1989/90, Exposé 718, Astérisque189–190 (1991), 107–131.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caniglia, L., Cortiñas, G., Danón, S. et al. Algorithmic aspects of Suslin's proof of Serre's conjecture. Comput Complexity 3, 31–55 (1993). https://doi.org/10.1007/BF01200406
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01200406
Key Words
- Serre's Conjecture
- Quillen-Suslin Theorem
- effective Nullstellensatz
- linear equation systems over polynomial rings
- complexity