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Volumn 75, Issue 256, 2006, Pages 2049-2085

Fast computation of a rational point of a variety over a finite field

Author keywords

First Bertini theorem; Geometric solutions; Probabilistic algorithms; Rational points; Straight line programs; Varieties over finite fields

Indexed keywords


EID: 33750880323     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-06-01878-3     Document Type: Article
Times cited : (17)

References (62)
  • 1
    • 0001906678 scopus 로고    scopus 로고
    • Zeroes, multiplicities and idempotents for zerodimensional systems
    • Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston), Birkhäuser, Boston. MR1414442 (97i:13027)
    • M.E. Alonso, E. Becker, M.-F. Roy, and T. Wörmann, Zeroes, multiplicities and idempotents for zerodimensional systems, Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston), Progr. Math., vol. 143, Birkhäuser, Boston, 1996, pp. 1-15. MR1414442 (97i:13027)
    • (1996) Progr. Math. , vol.143 , pp. 1-15
    • Alonso, M.E.1    Becker, E.2    Roy, M.-F.3    Wörmann, T.4
  • 2
    • 0031097237 scopus 로고    scopus 로고
    • Polar varieties and efficient real equation solving: The hypersurface case
    • MR1449757 (98h:68123)
    • B. Bank, M. Giusti, J. Heintz, and G.M. Mbakop, Polar varieties and efficient real equation solving: The hypersurface case, J. Complexity 13 (1997). no. 1, 5-27. MR1449757 (98h:68123)
    • (1997) J. Complexity , vol.13 , Issue.1 , pp. 5-27
    • Bank, B.1    Giusti, M.2    Heintz, J.3    Mbakop, G.M.4
  • 3
    • 0035465565 scopus 로고    scopus 로고
    • Polar varieties and efficient real elimination
    • MR1860738 (2002g:14084)
    • _, Polar varieties and efficient real elimination, Math. Z. 238 (2001), no. 1, 115-144. MR1860738 (2002g:14084)
    • (2001) Math. Z. , vol.238 , Issue.1 , pp. 115-144
  • 4
    • 12144249591 scopus 로고    scopus 로고
    • A first approach to generalized polar varieties
    • (Prague). MR2120995 (2006e:14078)
    • B. Bank, M. Giusti, J. Heintz, and L.M. Pardo, A first approach to generalized polar varieties, Kybernetika (Prague) 40 (2004), no. 5, 519-550. MR2120995 (2006e:14078)
    • (2004) Kybernetika , vol.40 , Issue.5 , pp. 519-550
    • Bank, B.1    Giusti, M.2    Heintz, J.3    Pardo, L.M.4
  • 5
    • 21844468940 scopus 로고    scopus 로고
    • Generalized polar varieties: Geometry and algorithms
    • MR2152713
    • _, Generalized polar varieties: Geometry and algorithms, J. Complexity 21 (2005), no. 4, 377-412. MR2152713
    • (2005) J. Complexity , vol.21 , Issue.4 , pp. 377-412
  • 6
    • 0003751005 scopus 로고
    • Polynomial and matrix computations
    • Birkhäuser, Boston. MR1289412 (95k:65003)
    • D. Bini and V. Pan, Polynomial and matrix computations, Progress in Theoretical Computer Science, Birkhäuser, Boston, 1994. MR1289412 (95k:65003)
    • (1994) Progress in Theoretical Computer Science
    • Bini, D.1    Pan, V.2
  • 7
    • 0000797215 scopus 로고
    • Time space tradeoffs (getting closer to the barriers?)
    • 4th International Symposium on Algorithms and Computation, ISAAC '93, Hong Kong, December 15-17, 1993 (Berlin), Springer
    • A. Borodin, Time space tradeoffs (getting closer to the barriers?), 4th International Symposium on Algorithms and Computation, ISAAC '93, Hong Kong, December 15-17, 1993 (Berlin), Lecture Notes in Comput. Sci., vol. 762, Springer, 1993, pp. 209-220.
    • (1993) Lecture Notes in Comput. Sci. , vol.762 , pp. 209-220
    • Borodin, A.1
  • 9
    • 33344475283 scopus 로고    scopus 로고
    • Improved explicit estimates on the number of solutions of equations over a finite field
    • A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12 (2006), no. 2, 155-185.
    • (2006) Finite Fields Appl. , vol.12 , Issue.2 , pp. 155-185
    • Cafure, A.1    Matera, G.2
  • 10
    • 0042461761 scopus 로고    scopus 로고
    • The hardness of polynomial equation solving
    • MR2009683 (2004k:68056)
    • D. Castro, M. Giusti, J. Heintz, G. Matera, and L.M. Pardo, The hardness of polynomial equation solving, Found. Comput. Math. 3 (2003), no. 4, 347-420. MR2009683 (2004k:68056)
    • (2003) Found. Comput. Math. , vol.3 , Issue.4 , pp. 347-420
    • Castro, D.1    Giusti, M.2    Heintz, J.3    Matera, G.4    Pardo, L.M.5
  • 12
    • 77649327985 scopus 로고    scopus 로고
    • Efficient algorithms for solving overdefined systems of multivariate polynomial equations
    • EUROCRYPT 2000 (Berlin) (B. Preneel, ed.), Springer. MR1772028
    • N. Courtois, A. Klimov, J. Patarin, and A. Shamir, Efficient algorithms for solving overdefined systems of multivariate polynomial equations, EUROCRYPT 2000 (Berlin) (B. Preneel, ed.), Lecture Notes in Comput. Sci., vol. 1807, Springer, 2000, pp. 71-79. MR1772028
    • (2000) Lecture Notes in Comput. Sci. , vol.1807 , pp. 71-79
    • Courtois, N.1    Klimov, A.2    Patarin, J.3    Shamir, A.4
  • 13
    • 0003908675 scopus 로고
    • Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra
    • Springer, New York. MR1189133 (93j:13031)
    • D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergrad. Texts Math., Springer, New York, 1992. MR1189133 (93j:13031)
    • (1992) Undergrad. Texts Math.
    • Cox, D.1    Little, J.2    O'Shea, D.3
  • 14
    • 0003352688 scopus 로고    scopus 로고
    • Using algebraic geometry
    • Springer, New York. MR1639811 (99h:13033)
    • _, Using algebraic geometry, Grad. Texts in Math., vol. 185, Springer, New York, 1998. MR1639811 (99h:13033)
    • (1998) Grad. Texts in Math. , vol.185
  • 15
    • 33750854289 scopus 로고    scopus 로고
    • Gröbner bases for codes
    • Some tapas in computer algebra (A. Cohen et al., ed.), Springer, Berlin. MR1679927 (2000d:94029a)
    • M. de Boer and R. Pellikaan, Gröbner bases for codes, Some tapas in computer algebra (A. Cohen et al., ed.), Algorithms Comput. Math., vol. 4, Springer, Berlin, 1999, pp. 237-259. MR1679927 (2000d:94029a)
    • (1999) Algorithms Comput. Math. , vol.4 , pp. 237-259
    • De Boer, M.1    Pellikaan, R.2
  • 16
    • 0003527787 scopus 로고
    • Commutative algebra with a view toward algebraic geometry
    • Springer, New York. MR1322960 (97a:13001)
    • D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math., vol. 150, Springer, New York, 1995. MR1322960 (97a:13001)
    • (1995) Grad. Texts in Math. , vol.150
    • Eisenbud, D.1
  • 17
    • 0036045901 scopus 로고    scopus 로고
    • A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
    • Lille, France, July 7-10, 2002 (New York) (T. Mora, ed.), ACM Press. MR2035234 (2005c:13033)
    • J.-C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F5), ISSAC'02: Proceedings of the International Symposium on Symbolic and Algebraic Computation, Lille, France, July 7-10, 2002 (New York) (T. Mora, ed.), ACM Press, 2002, pp. 75-83. MR2035234 (2005c:13033)
    • (2002) ISSAC'02: Proceedings of the International Symposium on Symbolic and Algebraic Computation , pp. 75-83
    • Faugère, J.-C.1
  • 18
    • 0004233493 scopus 로고
    • Springer, Berlin, Heidelberg, New York. MR0732620 (85k:14004)
    • W. Fulton, Intersection Theory, Springer, Berlin, Heidelberg, New York, 1984. MR0732620 (85k:14004)
    • (1984) Intersection Theory
    • Fulton, W.1
  • 19
    • 85034753633 scopus 로고
    • Algebraic solution of systems of polynomial equations using Gröbner bases
    • 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)
    • P. Gianni and T. Mora, Algebraic solution of systems of polynomial equations using Gröbner bases, 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.), Lecture Notes in Comput. Sci., vol. 356, Springer, 1989, pp. 247-257. MR1008541 (91e:13024)
    • (1989) Lecture Notes in Comput. Sci. , vol.356 , pp. 247-257
    • Gianni, P.1    Mora, T.2
  • 22
    • 84955611783 scopus 로고
    • When polynomial equation systems can be solved fast?
    • Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings AAECC-11 (Berlin) (G. Cohen, M. Giusti, and T. Mora, eds.), Springer. MR1448166 (98a:68106)
    • M. Giusti, J. Heintz, J.E. Morais, and L.M. Pardo, When polynomial equation systems can be solved fast?, Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings AAECC-11 (Berlin) (G. Cohen, M. Giusti, and T. Mora, eds.), Lecture Notes in Comput. Sci., vol. 948, Springer, 1995, pp. 205-231. MR1448166 (98a:68106)
    • (1995) Lecture Notes in Comput. Sci. , vol.948 , pp. 205-231
    • Giusti, M.1    Heintz, J.2    Morais, J.E.3    Pardo, L.M.4
  • 23
    • 0031539424 scopus 로고    scopus 로고
    • Le rôle des structures de données dans les problèmes d'élimination
    • MR1490129 (98j:68068)
    • _, Le rôle des structures de données dans les problèmes d'élimination, C. R. Math. Acad. Sci. Paris 325 (1997), 1223-1228. MR1490129 (98j:68068)
    • (1997) C. R. Math. Acad. Sci. Paris , vol.325 , pp. 1223-1228
  • 24
    • 0001905196 scopus 로고
    • On the efficiency of effective Nullstellensätze
    • MR1220078 (94i:13016)
    • M. Giusti, J. Heintz, and J. Sabia, On the efficiency of effective Nullstellensätze, Comput. Complexity 3 (1993), 56-95. MR1220078 (94i:13016)
    • (1993) Comput. Complexity , vol.3 , pp. 56-95
    • Giusti, M.1    Heintz, J.2    Sabia, J.3
  • 25
    • 0035294718 scopus 로고    scopus 로고
    • A Gröbner free alternative for polynomial system solving
    • MR1817612 (2002b:68123)
    • M. Giusti, G. Lecerf, and B. Salvy, A Gröbner free alternative for polynomial system solving, J. Complexity 17 (2001), no. 1, 154-211. MR1817612 (2002b:68123)
    • (2001) J. Complexity , vol.17 , Issue.1 , pp. 154-211
    • Giusti, M.1    Lecerf, G.2    Salvy, B.3
  • 26
    • 0000627865 scopus 로고
    • Definability and fast quantifier elimination in algebraically closed fields
    • MR0716823 (85a:68062)
    • J. Heintz, Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci. 24 (1983), no. 3, 239-277. MR0716823 (85a:68062)
    • (1983) Theoret. Comput. Sci. , vol.24 , Issue.3 , pp. 239-277
    • Heintz, J.1
  • 27
    • 0001365155 scopus 로고
    • On the computational complexity of polynomials and bilinear mappings. A survey
    • 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)
    • _, On the computational complexity of polynomials and bilinear mappings. A survey, 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.), Lecture Notes in Comput. Sci., vol. 356, Springer, 1989, pp. 269-300. MR1008524 (90d:94001)
    • (1989) Lecture Notes in Comput. Sci. , vol.356 , pp. 269-300
  • 28
    • 0002174190 scopus 로고    scopus 로고
    • The intrinsic complexity of parametric elimination methods
    • MR1675449 (2000b:65249)
    • J. Heintz, G. Matera, L.M. Pardo, and R. Wachenchauzer, The intrinsic complexity of parametric elimination methods, Electron. J. SADIO 1 (1998), no. 1, 37-51. MR1675449 (2000b:65249)
    • (1998) Electron. J. SADIO , vol.1 , Issue.1 , pp. 37-51
    • Heintz, J.1    Matera, G.2    Pardo, L.M.3    Wachenchauzer, R.4
  • 29
    • 0035060036 scopus 로고    scopus 로고
    • On the time-space complexity of geometric elimination procedures
    • MR1818975 (2002c:68108)
    • J. Heintz, G. Matera, and A. Waissbein, On the time-space complexity of geometric elimination procedures, Appl. Algebra Engrg. Comm. Comput. 11 (2001), no. 4, 239-296. MR1818975 (2002c:68108)
    • (2001) Appl. Algebra Engrg. Comm. Comput. , vol.11 , Issue.4 , pp. 239-296
    • Heintz, J.1    Matera, G.2    Waissbein, A.3
  • 30
    • 0033273463 scopus 로고    scopus 로고
    • Solvability of systems of polynomial congruences modulo a large prime
    • MR1737238 (2000j:11044)
    • M.-D. Huang and Y.-C. Wong, Solvability of systems of polynomial congruences modulo a large prime, Comput. Complexity 8 (1999), no. 3, 227-257. MR1737238 (2000j:11044)
    • (1999) Comput. Complexity , vol.8 , Issue.3 , pp. 227-257
    • Huang, M.-D.1    Wong, Y.-C.2
  • 31
    • 0348000553 scopus 로고    scopus 로고
    • Extended Hilbert irreducibility and its applications
    • MR1783251 (2001h:12002)
    • _, Extended Hilbert irreducibility and its applications, J. Algorithms 37 (2000), no. 1, 121-145. MR1783251 (2001h:12002)
    • (2000) J. Algorithms , vol.37 , Issue.1 , pp. 121-145
  • 32
    • 0029291305 scopus 로고
    • Effective Noether irreducibility forms and applications
    • MR1330258 (96g:68053)
    • E. Kaltofen, Effective Noether irreducibility forms and applications, J. Comput. System Sci. 50 (1995), no. 2, 274-295. MR1330258 (96g:68053)
    • (1995) J. Comput. System Sci. , vol.50 , Issue.2 , pp. 274-295
    • Kaltofen, E.1
  • 33
    • 84957060922 scopus 로고    scopus 로고
    • Cryptanalysis of the HFE PublicKeyCryptosystem by relinearization
    • Proceedings of Advances in Cryptology - CRYPTO'99, Santa Barbara, California, USA. August 15-19, 1999 (Berlin) (M.J. Wiener, ed.), Springer. MR1729291 (2000i:94052)
    • A. Kipnis and A. Shamir, Cryptanalysis of the HFE PublicKeyCryptosystem by relinearization, Proceedings of Advances in Cryptology - CRYPTO'99, Santa Barbara, California, USA. August 15-19, 1999 (Berlin) (M.J. Wiener, ed.), Lecture Notes in Comput. Sci., vol. 1666, Springer, 1999, pp. 19-30. MR1729291 (2000i:94052)
    • (1999) Lecture Notes in Comput. Sci. , vol.1666 , pp. 19-30
    • Kipnis, A.1    Shamir, A.2
  • 34
    • 0000697864 scopus 로고    scopus 로고
    • A computational method for diophantine approximation
    • Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston) (L. González-Vega and T. Recio, eds.), Birkhäuser Boston. MR1414452 (98h:13039)
    • T. Krick and L.M. Pardo, A computational method for Diophantine approximation, Algorithms in Algebraic Geometry and Applications, Proceedings of MEGA'94 (Boston) (L. González-Vega and T. Recio, eds.), Progr. Math., vol. 143, Birkhäuser Boston, 1996, pp. 193-254. MR1414452 (98h:13039)
    • (1996) Progr. Math. , vol.143 , pp. 193-254
    • Krick, T.1    Pardo, L.M.2
  • 35
    • 84941839155 scopus 로고
    • Grundzüge einer arithmetischen Theorie der algebraischen Grössen
    • L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Grössen, J. Reine Angew. Math. 92 (1882), 1-122.
    • (1882) J. Reine Angew. Math. , vol.92 , pp. 1-122
    • Kronecker, L.1
  • 37
    • 0041459777 scopus 로고    scopus 로고
    • Quadratic Newton iteration for systems with multiplicity
    • MR1907381 (2003f:65090)
    • G. Lecerf, Quadratic Newton iteration for systems with multiplicity, Found. Comput. Math. 2 (2002), no. 3, 247-293. MR1907381 (2003f:65090)
    • (2002) Found. Comput. Math. , vol.2 , Issue.3 , pp. 247-293
    • Lecerf, G.1
  • 38
    • 0042156945 scopus 로고    scopus 로고
    • Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers
    • MR1991984 (2004j:68200)
    • _, Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers, J. Complexity 19 (2003), no. 4, 564-596. MR1991984 (2004j:68200)
    • (2003) J. Complexity , vol.19 , Issue.4 , pp. 564-596
  • 39
    • 0004127220 scopus 로고
    • Addison Wesley, Reading. Massachusetts. MR0746963 (86c:11106)
    • R. Lidl and H. Niederreiter, Finite fields. Addison Wesley, Reading. Massachusetts, 1983. MR0746963 (86c:11106)
    • (1983) Finite Fields.
    • Lidl, R.1    Niederreiter, H.2
  • 40
    • 0004017469 scopus 로고
    • Applied abstract algebra
    • Springer, New York. MR0765220 (86d:00002)
    • R. Lidl and G. Pilz, Applied abstract algebra, Undergrad. Texts Math., Springer, New York. 1984. MR0765220 (86d:00002)
    • (1984) Undergrad. Texts Math.
    • Lidl, R.1    Pilz, G.2
  • 44
    • 0003521409 scopus 로고
    • Algebraic geometry I. Complex projective varieties
    • 2nd ed., Springer, Berlin. MR1344216 (96d:14001)
    • D. Mumford, Algebraic geometry I. Complex projective varieties, 2nd ed., Classics Math., Springer, Berlin, 1995. MR1344216 (96d:14001)
    • (1995) Classics Math.
    • Mumford, D.1
  • 45
    • 84955604612 scopus 로고
    • How lower and upper complexity bounds meet in elimination theory
    • 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)
    • L.M. Pardo, How lower and upper complexity bounds meet in elimination theory, Applied Algebra, Algebraic Algorithms and Error Correcting Codes, Proceedings of AAECC-11 (Berlin) (G. Cohen, M. Giusti, and T. Mora, eds.), Lecture Notes in Comput. Sci., vol. 948, Springer, 1995, pp. 33-69. MR1448154 (99a:68097)
    • (1995) Lecture Notes in Comput. Sci. , vol.948 , pp. 33-69
    • Pardo, L.M.1
  • 46
    • 0033131339 scopus 로고    scopus 로고
    • Solving zero-dimensional systems through rational univariate representation
    • MR1697179 (2000e:13038)
    • F. Rouillier, Solving zero-dimensional systems through rational univariate representation. Appl. Algebra Engrg. Comm. Comput. 9 (1997), no. 5, 433-461. MR1697179 (2000e:13038)
    • (1997) Appl. Algebra Engrg. Comm. Comput. , vol.9 , Issue.5 , pp. 433-461
    • Rouillier, F.1
  • 49
    • 0000237109 scopus 로고
    • A lower bound for the number of solutions of equations over finite fields
    • MR0360598 (50:13045)
    • W. Schmidt, A lower bound for the number of solutions of equations over finite fields, J. Number Theory 6 (1974), no. 6, 448-480. MR0360598 (50:13045)
    • (1974) J. Number Theory , vol.6 , Issue.6 , pp. 448-480
    • Schmidt, W.1
  • 50
    • 0004034096 scopus 로고
    • Equations over finite fields. An elementary approach
    • Springer, New York. MR0429733 (55:2744)
    • _, Equations over finite fields. An elementary approach, Lectures Notes in Math., no. 536, Springer, New York, 1976. MR0429733 (55:2744)
    • (1976) Lectures Notes in Math. , Issue.536
  • 51
    • 0037332265 scopus 로고    scopus 로고
    • Computing parametric geometric resolutions
    • MR1959170 (2003k:13035)
    • E. Schost, Computing parametric geometric resolutions, Appl. Algebra Engrg. Comm. Comput. 13 (2003), 349-393. MR1959170 (2003k:13035)
    • (2003) Appl. Algebra Engrg. Comm. Comput. , vol.13 , pp. 349-393
    • Schost, E.1
  • 52
    • 84976779342 scopus 로고
    • Fast probabilistic algorithms for verification of polynomial identities
    • MR0594695 (82m:68078)
    • J.T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. ACM 27 (1980), no. 4, 701-717. MR0594695 (82m:68078)
    • (1980) J. ACM , vol.27 , Issue.4 , pp. 701-717
    • Schwartz, J.T.1
  • 53
    • 0004287752 scopus 로고
    • Basic algebraic geometry
    • Springer, New York. MR0447223 (56:5538)
    • I.R. Shafarevich, Basic algebraic geometry, Grad. Texts in Math., Springer, New York, 1984. MR0447223 (56:5538)
    • (1984) Grad. Texts in Math.
    • Shafarevich, I.R.1
  • 55
    • 0001052336 scopus 로고
    • Algebraic complexity theory
    • (J. van Leeuwen, ed.), Elsevier, Amsterdam. MR1127177
    • V. Strassen, Algebraic complexity theory, Handbook of Theoretical Computer Science (J. van Leeuwen, ed.), Elsevier, Amsterdam, 1990, pp. 634-671. MR1127177
    • (1990) Handbook of Theoretical Computer Science , pp. 634-671
    • Strassen, V.1
  • 56
    • 84957621752 scopus 로고
    • Parallel arithmetic computations: A survey
    • 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
    • J. von zur Gathen, Parallel arithmetic computations: a survey, 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.), Lecture Notes in Comput. Sci., vol. 233, Springer, August 1986, pp. 93-112. MR0874591
    • (1986) Lecture Notes in Comput. Sci. , vol.233 , pp. 93-112
    • Von Zur Gathen, J.1
  • 58
    • 0040007950 scopus 로고    scopus 로고
    • Counting curves and their projections
    • MR1436303 (98d:68111)
    • J. von zur Gathen, M. Karpinski, and I. Shparlinski, Counting curves and their projections, Comput. Complexity 6 (1997), no. 3, 64-99. MR1436303 (98d:68111)
    • (1997) Comput. Complexity , vol.6 , Issue.3 , pp. 64-99
    • Von Zur Gathen, J.1    Karpinski, M.2    Shparlinski, I.3
  • 59
    • 0942288265 scopus 로고    scopus 로고
    • Finding points on curves over finite fields
    • MR2034245 (2005b:68293)
    • J. von zur Gathen, I. Shparlinski, and A. Sinclair, Finding points on curves over finite fields, SIAM J. Comput. 32 (2003), no. 6, 1436-1448. MR2034245 (2005b:68293)
    • (2003) SIAM J. Comput. , vol.32 , Issue.6 , pp. 1436-1448
    • Von Zur Gathen, J.1    Shparlinski, I.2    Sinclair, A.3
  • 61
    • 0004105164 scopus 로고
    • Algebraic surfaces
    • Springer, Berlin. MR1336146 (96c:14024)
    • O. Zariski, Algebraic surfaces, Classics Math., Springer, Berlin, 1995. MR1336146 (96c:14024)
    • (1995) Classics Math.
    • Zariski, O.1
  • 62
    • 84925463455 scopus 로고
    • Probabilistic algorithms for sparse polynomials
    • EUROSAM '79: Proceedings of International Symposium on Symbolic and Algebraic Computation, Marseille 1979 (Berlin), Springer. MR0575692 (81g:68061)
    • R. Zippel, Probabilistic algorithms for sparse polynomials, EUROSAM '79: Proceedings of International Symposium on Symbolic and Algebraic Computation, Marseille 1979 (Berlin), Lecture Notes in Comput. Sci., vol. 72, Springer, 1979, pp. 216-226. MR0575692 (81g:68061)
    • (1979) Lecture Notes in Comput. Sci. , vol.72 , pp. 216-226
    • Zippel, R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.