Solvability of systems of polynomial congruences modulo a large prime

Computational Complexity

Volumn 8, Issue 3, 1999, Pages 227-257

  Huang, Ming Deh ^a   Wong, Yiu Chung ^a,b  

^a University of Southern California   (United States)

^b SYNOPSYS INC   (United States)

Author keywords

Algebraic set; Algorithm; Finite field; Polynomial congruence

Indexed keywords

EID: 0033273463     PISSN: 10163328     EISSN: None     Source Type: Journal    
DOI: 10.1007/s000370050029     Document Type: Article

References (28)

1. M. BEN-OR AND P. TIWARI, A deterministic algorithm for sparse multivariate polynomial interpolation. In Proceedings of the 20th ACM Symposium on the Theory of Computing. The Association of Computing Machinery, 1988, 301-309.
2. S. BERKOWITZ, On computing the determinant in small parallel time using a small number of processors. Information Processing Letters 18 (1984), 147-150.
3. A. BORODIN, J. VON ZUR GATHEN, AND J. E. HOPCROFT, Fast parallel matrix and GCD computations. Information and Control 52 (1982), 241-256.
4. A. L. CHISTOV, Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time. Journal of Soviet Mathematics 34(4) (1986), 1838-1882.
5. A. L. CHISTOV AND D. YU. GRIGOR'EV, Subexponential-time solving systems of algebraic equations, II. Preprint E-5-82, LOMI, Leningrad, 1983.
6. L. CSANKY, Fast parallel matrix inversion algorithms. SIAM Journal of Computing 5 (1976), 618-623.
7. JOACHIM VON ZUR GATHEN AND ERICH KALTOFEN, A polynomial-time algorithm for factoring multivariate polynomials over finite fields. In Proceedings of 1983 International Conference on Automata, Languages, and Programming, Lecture Notes in Computer Science 154, Springer Verlag, 1983, 250-263.
8. JOACHIM VON ZUR GATHEN AND IGOR SHPARLINSKI, Finding points on curves over finite fields. In Proceedings of 36th IEEE Symposium on Foundation of Computer Science, 1995.
9. JOACHIM VON ZUR GATHEN, MAREK KARPINSKI, AND IGOR SHPARLINSKI, Counting curves and their projections. Computational Complexity 6 (1997), 64-99.
10. MARC GIUSTI AND JOOS HEINTZ, Algorithmes - disons rapides - pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionnelles. In Effective Methods in Algebraic Geometry, ed. TEO MORA AND CARLO TRAVERSO, vol. 94 of Progress in Mathematics, 169-194. Birkhäuser, 1991.
11. MARC GIUSTI AND JOOS HEINTZ, La détermination des points isolés et de la dimension d'une variété algébrique peut se faire en temps polynomial. In Computational Algebraic Geometry and Commutative Algebra, ed. D. EISENBUD AND L. ROBBIANO. Cambridge University Press, 1993.
12. D. YU. GRIGOR'EV, Factorization of polynomials over a finite field and the solution of systems of algebraic equations. Journal of Soviet Mathematics 34(4) (1986), 1762-1803.
13. DIMA GRIGORIEV AND MAREK KARPINSKI, An approximation algorithm for the number of zeros of arbitrary polynomial over GF[q]. In Proceedings of 32nd IEEE Symposium on Foundation of Computer Science, 1991, 662-669.
14. MING-DEH HUANG AND DOUG IERARDI, Counting rational points on curves over finite fields. In Proceedings of 34th IEEE Symposium on Foundation of Computer Science. IEEE, 1993, 616-625.
15. MING-DEH HUANG AND ASHWIN RAO, Interpolation of sparse multivariate polynomials over finite fields with applications. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 1996, 508-517.
16. MING-DEH HUANG AND YIU-CHUNG WONG, An algorithm for approximate counting of points on an algebraic set over large finite fields. In Algorithmic Number Theory, Proceedings of the Third International Symposium (ANTS-III), Portland OR. Springer, 1998, 514-527.
17. DOUG IERARDI, Quantifier elimination in the theory of an algebraically closed field. In Proceedings of 21st ACM Symposium on Theory of Computation, 1989a.
18. DOUGLAS JOHN IERARDI, The Complexity of Quantifier Elimination in the Theory of an Algebraically Closed Field. PhD thesis, Department of Computer Science, Cornell University, Ithaca, New York 14853-7501, 1989b. Also available as Technical Report no. TR 89-1030 of Computer Science Department, Cornell University.
19. DOUGLAS IERARDI AND DEXTER KOZEN, Parallel resultant computation. In Synthesis of Parallel Algorithms. Kauffman, 1991.
20. ERICH KALTOFEN, Fast parallel absolute irreducibility testing. Journal of Symbolic Computation 1 (1985), 57-67.
21. SERGE LANG, Introduction to Algebraic Geometry, Section II.5. Addison-Wesley Publishing, Reading, Massachusetts, 1973.
22. K. MULMULEY, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7(1) (1987), 101-104.
23. MICHAEL O. RABIN, Probabilistic algorithms in finite fields. SIAM Journal of Computing 9 (1980), 273-280.
24. J. RENEGAR, On the computatioinal complexity and geometry of the first order theory of the reals, parts i-iii. Journal of Symbolic Computing 13(3) (1992), 255-352.
25. WOLFGANG M. SCHMIDT, A lower bound for the number of solutions of equations over finite fields. Journal of Number Theory 6 (1974), 448-480.
26. DOUGLAS H. WIEDEMANN, Solving sparse linear equations over finite fields. IEEE Transactions on Information Theory IT-32(1) (1986), 54-62.
27. YIU-CHUNG WONG, Algebraic Sets Decomposition, Polynomial Congruences, and Extended Hilbert Irreducibility. Ph.D thesis, University of Southern California, 1997.
28. OSCAR ZARISKI AND PIERRE SAMUEL, Commutative Algebra, vol. Vol 1. D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958.