Artículo

Abstract:

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety. © 2006 American Mathematical Society.

Registro:

Documento: Artículo
Título:Fast computation of a rational point of a variety over a finite field
Autor:Cafure, A.; Matera, G.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Pabellón I, (1428) Buenos Aires, Argentina
Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina
National Council of Science and Technology (CONICET), Argentina
Palabras clave:First Bertini theorem; Geometric solutions; Probabilistic algorithms; Rational points; Straight-line programs; Varieties over finite fields
Año:2006
Volumen:75
Número:256
Página de inicio:2049
Página de fin:2085
DOI: http://dx.doi.org/10.1090/S0025-5718-06-01878-3
Título revista:Mathematics of Computation
Título revista abreviado:Math. Comput.
ISSN:00255718
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00255718_v75_n256_p2049_Cafure.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00255718_v75_n256_p2049_Cafure

Referencias:

  • Alonso, M.E., Becker, E., Roy, M.-F., Wörmann, T., Zeroes, multiplicities and idempotents for zerodimensional systems (1996) Progr. Math., 143, pp. 1-15. , Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston), Birkhäuser, Boston. MR1414442 (97i:13027)
  • Bank, B., Giusti, M., Heintz, J., Mbakop, G.M., Polar varieties and efficient real equation solving: The hypersurface case (1997) J. Complexity, 13 (1), pp. 5-27. , MR1449757 (98h:68123)
  • Polar varieties and efficient real elimination (2001) Math. Z., 238 (1), pp. 115-144. , MR1860738 (2002g:14084)
  • Bank, B., Giusti, M., Heintz, J., Pardo, L.M., A first approach to generalized polar varieties (2004) Kybernetika, 40 (5), pp. 519-550. , (Prague). MR2120995 (2006e:14078)
  • Generalized polar varieties: Geometry and algorithms (2005) J. Complexity, 21 (4), pp. 377-412. , MR2152713
  • Bini, D., Pan, V., Polynomial and matrix computations (1994) Progress in Theoretical Computer Science, , Birkhäuser, Boston. MR1289412 (95k:65003)
  • Borodin, A., Time space tradeoffs (getting closer to the barriers?) (1993) Lecture Notes in Comput. Sci., 762, pp. 209-220. , 4th International Symposium on Algorithms and Computation, ISAAC '93, Hong Kong, December 15-17, 1993 (Berlin), Springer
  • Bürgisser, P., Clausen, M., Shokrollahi, M.A., Algebraic complexity theory (1997) Grundlehren Math. Wiss., 315. , Springer, Berlin. MR1440179 (99c:68002)
  • Cafure, A., Matera, G., Improved explicit estimates on the number of solutions of equations over a finite field (2006) Finite Fields Appl., 12 (2), pp. 155-185
  • Castro, D., Giusti, M., Heintz, J., Matera, G., Pardo, L.M., The hardness of polynomial equation solving (2003) Found. Comput. Math., 3 (4), pp. 347-420. , MR2009683 (2004k:68056)
  • Chistov, A.L., Grigoriev, D.Y., (1983) Subexponential Time Solving Systems of Algebraic Equations. I, II, , LOMI preprints E-9-83, E-10-83, Steklov Institute, Leningrad
  • Courtois, N., Klimov, A., Patarin, J., Shamir, A., Efficient algorithms for solving overdefined systems of multivariate polynomial equations (2000) Lecture Notes in Comput. Sci., 1807, pp. 71-79. , EUROCRYPT 2000 (Berlin) (B. Preneel, ed.), Springer. MR1772028
  • Cox, D., Little, J., O'Shea, D., Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra (1992) Undergrad. Texts Math., , Springer, New York. MR1189133 (93j:13031)
  • Using algebraic geometry (1998) Grad. Texts in Math., 185. , Springer, New York. MR1639811 (99h:13033)
  • De Boer, M., Pellikaan, R., Gröbner bases for codes (1999) Algorithms Comput. Math., 4, pp. 237-259. , Some tapas in computer algebra (A. Cohen et al., ed.), Springer, Berlin. MR1679927 (2000d:94029a)
  • Eisenbud, D., Commutative algebra with a view toward algebraic geometry (1995) Grad. Texts in Math., 150. , Springer, New York. MR1322960 (97a:13001)
  • Faugère, J.-C., A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) (2002) ISSAC'02: Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 75-83. , Lille, France, July 7-10, 2002 (New York) (T. Mora, ed.), ACM Press. MR2035234 (2005c:13033)
  • Fulton, W., (1984) Intersection Theory, , Springer, Berlin, Heidelberg, New York. MR0732620 (85k:14004)
  • Gianni, P., Mora, T., Algebraic solution of systems of polynomial equations using Gröbner bases (1989) Lecture Notes in Comput. Sci., 356, pp. 247-257. , Proceedings 5th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-5, Menorca, Spain, June 15-19, 1987 (Berlin) (L. Huguet and A. Poli, eds.), Springer. MR1008541 (91e:13024)
  • Giusti, M., Hägele, K., Heintz, J., Morais, J.E., Montana, J.L., Pardo, L.M., Lower bounds for diophantine approximation (1997) J. Pure Appl. Algebra, 117-118, pp. 277-317. , MR1457843 (99d:68106)
  • Giusti, M., Heintz, J., Morais, J.E., Morgenstern, J., Pardo, L.M., Straight-line programs in geometric elimination theory (1998) J. Pure Appl. Algebra, 124, pp. 101-146. , MR1600277 (99d:68128)
  • Giusti, M., Heintz, J., Morais, J.E., Pardo, L.M., When polynomial equation systems can be solved fast? (1995) Lecture Notes in Comput. Sci., 948, pp. 205-231. , Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings AAECC-11 (Berlin) (G. Cohen, M. Giusti, and T. Mora, eds.), Springer. MR1448166 (98a:68106)
  • Le rôle des structures de données dans les problèmes d'élimination (1997) C. R. Math. Acad. Sci. Paris, 325, pp. 1223-1228. , MR1490129 (98j:68068)
  • Giusti, M., Heintz, J., Sabia, J., On the efficiency of effective Nullstellensätze (1993) Comput. Complexity, 3, pp. 56-95. , MR1220078 (94i:13016)
  • Giusti, M., Lecerf, G., Salvy, B., A Gröbner free alternative for polynomial system solving (2001) J. Complexity, 17 (1), pp. 154-211. , MR1817612 (2002b:68123)
  • Heintz, J., Definability and fast quantifier elimination in algebraically closed fields (1983) Theoret. Comput. Sci., 24 (3), pp. 239-277. , MR0716823 (85a:68062)
  • On the computational complexity of polynomials and bilinear mappings. A survey (1989) Lecture Notes in Comput. Sci., 356, pp. 269-300. , Proceedings 5th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-5, Menorca, Spain, June 15-19, 1987 (Berlin) (L. Huguet and A. Poli, eds.), Springer. MR1008524 (90d:94001)
  • Heintz, J., Matera, G., Pardo, L.M., Wachenchauzer, R., The intrinsic complexity of parametric elimination methods (1998) Electron. J. SADIO, 1 (1), pp. 37-51. , MR1675449 (2000b:65249)
  • Heintz, J., Matera, G., Waissbein, A., On the time-space complexity of geometric elimination procedures (2001) Appl. Algebra Engrg. Comm. Comput., 11 (4), pp. 239-296. , MR1818975 (2002c:68108)
  • Huang, M.-D., Wong, Y.-C., Solvability of systems of polynomial congruences modulo a large prime (1999) Comput. Complexity, 8 (3), pp. 227-257. , MR1737238 (2000j:11044)
  • Extended Hilbert irreducibility and its applications (2000) J. Algorithms, 37 (1), pp. 121-145. , MR1783251 (2001h:12002)
  • Kaltofen, E., Effective Noether irreducibility forms and applications (1995) J. Comput. System Sci., 50 (2), pp. 274-295. , MR1330258 (96g:68053)
  • Kipnis, A., Shamir, A., Cryptanalysis of the HFE PublicKeyCryptosystem by relinearization (1999) Lecture Notes in Comput. Sci., 1666, pp. 19-30. , Proceedings of Advances in Cryptology - CRYPTO'99, Santa Barbara, California, USA. August 15-19, 1999 (Berlin) (M.J. Wiener, ed.), Springer. MR1729291 (2000i:94052)
  • Krick, T., Pardo, L.M., A computational method for diophantine approximation (1996) Progr. Math., 143, pp. 193-254. , Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston) (L. González-Vega and T. Recio, eds.), Birkhäuser Boston. MR1414452 (98h:13039)
  • Kronecker, L., Grundzüge einer arithmetischen Theorie der algebraischen Grössen (1882) J. Reine Angew. Math., 92, pp. 1-122
  • Kunz, E., (1985) Introduction to Commutative Algebra and Algebraic Geometry, , Birkhäuser, Boston. MR0789602 (86e:14001)
  • Lecerf, G., Quadratic Newton iteration for systems with multiplicity (2002) Found. Comput. Math., 2 (3), pp. 247-293. , MR1907381 (2003f:65090)
  • Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers (2003) J. Complexity, 19 (4), pp. 564-596. , MR1991984 (2004j:68200)
  • Lidl, R., Niederreiter, H., (1983) Finite Fields., , Addison Wesley, Reading. Massachusetts. MR0746963 (86c:11106)
  • Lidl, R., Pilz, G., Applied abstract algebra (1984) Undergrad. Texts Math., , Springer, New York. MR0765220 (86d:00002)
  • Macaulay, F.S., (1916) The Algebraic Theory of Modular Systems, , Cambridge Univ. Press, Cambridge. MR1281612 (951:13001)
  • Matsumura, H., (1980) Commutative Algebra, , Benjamin. MR0575344 (82i:13003)
  • Morais, J.E., (1997) Resolución Eficaz de Sistemas de Ecuaciones Polinomiales, , Ph.D. thesis, Universidad de Cantabria, Santander, Spain
  • Mumford, D., Algebraic geometry I. Complex projective varieties (1995) Classics Math., , 2nd ed., Springer, Berlin. MR1344216 (96d:14001)
  • Pardo, L.M., How lower and upper complexity bounds meet in elimination theory (1995) Lecture Notes in Comput. Sci., 948, pp. 33-69. , Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings of AAECC-11 (Berlin) (G. Cohen, M. Giusti, and T. Mora, eds.), Springer. MR1448154 (99a:68097)
  • Rouillier, F., Solving zero-dimensional systems through rational univariate representation (1997) Appl. Algebra Engrg. Comm. Comput., 9 (5), pp. 433-461. , MR1697179 (2000e:13038)
  • Samuel, P., (1967) Méthodes d'Algèbre Abstraite en Géométrie Algébrique, , Springer, Berlin, Heidelberg, New York. MR0213347 (35:4211)
  • Savage, J.E., (1998) Models of Computation. Exploring the Power of Computing, , Addison-Wesley, Reading, Massachussets
  • Schmidt, W., A lower bound for the number of solutions of equations over finite fields (1974) J. Number Theory, 6 (6), pp. 448-480. , MR0360598 (50:13045)
  • Equations over finite fields. An elementary approach (1976) Lectures Notes in Math., (536). , Springer, New York. MR0429733 (55:2744)
  • Schost, E., Computing parametric geometric resolutions (2003) Appl. Algebra Engrg. Comm. Comput., 13, pp. 349-393. , MR1959170 (2003k:13035)
  • Schwartz, J.T., Fast probabilistic algorithms for verification of polynomial identities (1980) J. ACM, 27 (4), pp. 701-717. , MR0594695 (82m:68078)
  • Shafarevich, I.R., Basic algebraic geometry (1984) Grad. Texts in Math., , Springer, New York. MR0447223 (56:5538)
  • (1994) Basic Algebraic Geometry: Varieties in Projective Space, , Springer, Berlin, Heidelberg, New York. MR1328833 (95m:14001)
  • Strassen, V., Algebraic complexity theory (1990) Handbook of Theoretical Computer Science, pp. 634-671. , (J. van Leeuwen, ed.), Elsevier, Amsterdam. MR1127177
  • Von Zur Gathen, J., Parallel arithmetic computations: A survey (1986) Lecture Notes in Comput. Sci., 233, pp. 93-112. , Proceedings of the 12th International Symposium on Mathematical Foundations of Computer Science, Bratislava, Czechoslovakia, August 25-29, 1996 (Berlin) (J. Gruska, B. Rovan, and J. Wiedermann, eds.), Springer, August. MR0874591
  • Von Zur Gathen, J., Gerhard, J., (1999) Modern Computer Algebra, , Cambridge Univ. Press, Cambridge. MR1689167 (2000j:68205)
  • Von Zur Gathen, J., Karpinski, M., Shparlinski, I., Counting curves and their projections (1997) Comput. Complexity, 6 (3), pp. 64-99. , MR1436303 (98d:68111)
  • Von Zur Gathen, J., Shparlinski, I., Sinclair, A., Finding points on curves over finite fields (2003) SIAM J. Comput., 32 (6), pp. 1436-1448. , MR2034245 (2005b:68293)
  • Weil, A., (1948) Sur les Courbes Algébriques et les Variétés qui s'en Déduisent, , Hermann, Paris. MR0027151 (10:262c)
  • Zariski, O., Algebraic surfaces (1995) Classics Math., , Springer, Berlin. MR1336146 (96c:14024)
  • Zippel, R., Probabilistic algorithms for sparse polynomials (1979) Lecture Notes in Comput. Sci., 72, pp. 216-226. , EUROSAM '79: Proceedings of International Symposium on Symbolic and Algebraic Computation, Marseille 1979 (Berlin), Springer. MR0575692 (81g:68061)

Citas:

---------- APA ----------
Cafure, A. & Matera, G. (2006) . Fast computation of a rational point of a variety over a finite field. Mathematics of Computation, 75(256), 2049-2085.
http://dx.doi.org/10.1090/S0025-5718-06-01878-3
---------- CHICAGO ----------
Cafure, A., Matera, G. "Fast computation of a rational point of a variety over a finite field" . Mathematics of Computation 75, no. 256 (2006) : 2049-2085.
http://dx.doi.org/10.1090/S0025-5718-06-01878-3
---------- MLA ----------
Cafure, A., Matera, G. "Fast computation of a rational point of a variety over a finite field" . Mathematics of Computation, vol. 75, no. 256, 2006, pp. 2049-2085.
http://dx.doi.org/10.1090/S0025-5718-06-01878-3
---------- VANCOUVER ----------
Cafure, A., Matera, G. Fast computation of a rational point of a variety over a finite field. Math. Comput. 2006;75(256):2049-2085.
http://dx.doi.org/10.1090/S0025-5718-06-01878-3