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Volumn 12, Issue 2, 2006, Pages 155-185

Improved explicit estimates on the number of solutions of equations over a finite field

Author keywords

Effective elimination theory; Effective first Bertini theorem; q rational points; Varieties over finite fields

Indexed keywords

PROBLEM SOLVING; THEOREM PROVING;

EID: 33344475283     PISSN: 10715797     EISSN: 10902465     Source Type: Journal    
DOI: 10.1016/j.ffa.2005.03.003     Document Type: Article
Times cited : (123)

References (27)
  • 1
    • 0001906678 scopus 로고    scopus 로고
    • Zeroes, multiplicities and idempotents for zerodimensional systems
    • [ABRW96], Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 Birkhäuser, Boston
    • [ABRW96] M.E. Alonso, E. Becker, M.-F. Roy, T. Wörmann, Zeroes, multiplicities and idempotents for zerodimensional systems, in: Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94, Progress in Mathematics, vol. 143, Birkhäuser, Boston, 1996, pp. 1-15.
    • (1996) Progress in Mathematics , vol.143 , pp. 1-15
    • Alonso, M.E.1    Becker, E.2    Roy, M.-F.3    Wörmann, T.4
  • 2
    • 0007467925 scopus 로고
    • Counting points on curves over finite fields
    • [Bom74], Exp 430, in: Séminaire Bourbaki 1972/1973 Springer, New York
    • [Bom74] E. Bombieri, Counting points on curves over finite fields (d'après S.A. Stepanov), Exp 430, in: Séminaire Bourbaki 1972/1973, Lecture Notes in Mathematics, vol. 383, Springer, New York, 1974, pp. 234-241.
    • (1974) Lecture Notes in Mathematics , vol.383 , pp. 234-241
    • Bombieri, E.1
  • 3
    • 33344479503 scopus 로고    scopus 로고
    • Explicit estimates for the number of solutions of polynomial equation systems over finite fields
    • [CM02], P. D'Argenio, G. Matera (Eds.), Proceedings Workshop Argentino de Informática Teórica, WAIT'02, Santa Fe, Argentina, September 2002 SADIO Buenos Aires
    • [CM02] A. Cafure, G. Matera, Explicit estimates for the number of solutions of polynomial equation systems over finite fields, in: P. D'Argenio, G. Matera (Eds.), Proceedings Workshop Argentino de Informática Teórica, WAIT'02, Santa Fe, Argentina, September 2002, Anales Jornadas Argentinas de Informática e Investigación Operativa, vol. 31, SADIO, 2002, Buenos Aires, pp. 26-41.
    • (2002) Anales Jornadas Argentinas de Informática e Investigación Operativa , vol.31 , pp. 26-41
    • Cafure, A.1    Matera, G.2
  • 4
    • 33344478955 scopus 로고    scopus 로고
    • Fast computation of a rational point of a variety over a finite field
    • [CM02], Sarmiento
    • [CM02] A. Cafure, G. Matera, Fast computation of a rational point of a variety over a finite field, Manuscript Universidad Nacional de General. Sarmiento, 2003. Available at http://arxiv.org/abs/math.NT/0406085.
    • (2003) Manuscript Universidad Nacional de General
    • Cafure, A.1    Matera, G.2
  • 6
    • 0004233493 scopus 로고
    • Springer, Berlin Heidelberg, New York
    • W. Fulton Intersection Theory 1984 Springer, Berlin Heidelberg, New York
    • (1984) Intersection Theory
    • Fulton, W.1
  • 7
    • 0037376559 scopus 로고    scopus 로고
    • Factoring multivariate polynomials via partial differential equations
    • S. Gao Factoring multivariate polynomials via partial differential equations Math. Comp. 72 2003 801 822
    • (2003) Math. Comp. , vol.72 , pp. 801-822
    • Gao, S.1
  • 8
    • 0344774146 scopus 로고    scopus 로고
    • Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields
    • S. Ghorpade, and G. Lachaud Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields Moscow Math. J. 2 3 2002 589 631
    • (2002) Moscow Math. J. , vol.2 , Issue.3 , pp. 589-631
    • Ghorpade, S.1    Lachaud, G.2
  • 9
    • 33344475623 scopus 로고    scopus 로고
    • Number of solutions of equations over finite fields and a conjecture of Lang and Weil
    • A.K. Agarwal Hindustan Book Agency New Delhi*et al.
    • S. Ghorpade, and G. Lachaud Number of solutions of equations over finite fields and a conjecture of Lang and Weil A.K. Agarwal Number Theory and Discrete Mathematics (Chandigarh, 2000) 2002 Hindustan Book Agency New Delhi 269 291
    • (2002) Number Theory and Discrete Mathematics (Chandigarh, 2000) , pp. 269-291
    • Ghorpade, S.1    Lachaud, G.2
  • 10
    • 0000627865 scopus 로고
    • Definability and fast quantifier elimination in algebraically closed fields
    • J. Heintz Definability and fast quantifier elimination in algebraically closed fields Theoret. Comput. Sci. 24 3 1983 239 277
    • (1983) Theoret. Comput. Sci. , vol.24 , Issue.3 , pp. 239-277
    • Heintz, J.1
  • 11
    • 0001404855 scopus 로고
    • Testing polynomials which are easy to compute
    • [HS82], International Symposium on Logic and Algorithmic, Zurich, 1980
    • [HS82] J. Heintz, C.P. Schnorr, Testing polynomials which are easy to compute, in: International Symposium on Logic and Algorithmic, Zurich 1980, Monographies de L'Enseignement Mathematiques, vol. 30, 1982, pp. 237-254.
    • (1982) Monographies de l'Enseignement Mathematiques , vol.30 , pp. 237-254
    • Heintz, J.1    Schnorr, C.P.2
  • 12
    • 33947640477 scopus 로고    scopus 로고
    • An algorithm for approximate counting of points on algebraic sets over finite fields
    • [HW98], J. Buhler (Ed.), Third International Symposium on Algorithmic Number Theory, ANTS-III, Portland, Oregon, USA, June 21-25, 1998 Springer, Berlin
    • [HW98] M.-D. Huang, Y.-C. Wong, An algorithm for approximate counting of points on algebraic sets over finite fields, in: J. Buhler (Ed.), Third International Symposium on Algorithmic Number Theory, ANTS-III, Portland, Oregon, USA, June 21-25, 1998, Lecture Notes in Computer Science, vol. 1423, Springer, Berlin, 1998, pp. 514-527.
    • (1998) Lecture Notes in Computer Science , vol.1423 , pp. 514-527
    • Huang, M.-D.1    Wong, Y.-C.2
  • 13
    • 0033273463 scopus 로고    scopus 로고
    • Solvability of systems of polynomial congruences modulo a large prime
    • M.-D. Huang, and Y.-C. Wong Solvability of systems of polynomial congruences modulo a large prime Comput. Complexity 8 1999 227 257
    • (1999) Comput. Complexity , vol.8 , pp. 227-257
    • Huang, M.-D.1    Wong, Y.-C.2
  • 14
    • 0029291305 scopus 로고
    • Effective Noether irreducibility forms and applications
    • E. Kaltofen Effective Noether irreducibility forms and applications J. Comput. System Sci. 50 2 1995 274 295
    • (1995) J. Comput. System Sci. , vol.50 , Issue.2 , pp. 274-295
    • Kaltofen, E.1
  • 15
    • 0001555828 scopus 로고    scopus 로고
    • Sharp estimates for the Arithmetic Nullstellensatz
    • T. Krick, L.M. Pardo, and M. Sombra Sharp estimates for the Arithmetic Nullstellensatz Duke Math. J. 109 3 2001 521 598
    • (2001) Duke Math. J. , vol.109 , Issue.3 , pp. 521-598
    • Krick, T.1    Pardo, L.M.2    Sombra, M.3
  • 18
    • 0001018621 scopus 로고
    • The number of points of varieties in finite fields
    • S. Lang, and A. Weil The number of points of varieties in finite fields Amer. J. Math. 76 1954 819 827
    • (1954) Amer. J. Math. , vol.76 , pp. 819-827
    • Lang, S.1    Weil, A.2
  • 20
    • 0033131339 scopus 로고    scopus 로고
    • Solving zero-dimensional systems through rational univariate representation
    • F. Rouillier Solving zero-dimensional systems through rational univariate representation Appl. Algebra Eng. Comm. Comput. 9 5 1997 433 461
    • (1997) Appl. Algebra Eng. Comm. Comput. , vol.9 , Issue.5 , pp. 433-461
    • Rouillier, F.1
  • 21
    • 30744466015 scopus 로고
    • Zur Methode von Stepanov
    • W. Schmidt Zur Methode von Stepanov Acta Arith. 24 1973 347 367
    • (1973) Acta Arith. , vol.24 , pp. 347-367
    • Schmidt, W.1
  • 22
    • 0000237109 scopus 로고
    • A lower bound for the number of solutions of equations over finite fields
    • W. Schmidt A lower bound for the number of solutions of equations over finite fields J. Number Theory 6 6 1974 448 480
    • (1974) J. Number Theory , vol.6 , Issue.6 , pp. 448-480
    • Schmidt, W.1
  • 23
    • 0003215121 scopus 로고
    • Equations over finite fields: An elementary approach
    • [Sch76], Springer, New York
    • [Sch76] W. Schmidt, Equations over Finite Fields: An Elementary Approach, Lectures Notes in Mathematics, vol. 536, Springer, New York, 1976.
    • (1976) Lectures Notes in Mathematics , vol.536
    • Schmidt, W.1
  • 26
    • 0040269398 scopus 로고
    • On the number of points of a hyperelliptic curve over a finite prime field
    • S. Stepanov On the number of points of a hyperelliptic curve over a finite prime field Math. USSR Izv. 3 1971 1103 1114
    • (1971) Math. USSR Izv. , vol.3 , pp. 1103-1114
    • Stepanov, S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.