Proceedings of 37th Conference on Foundations of Computer Science (1996)
Oct. 14, 1996 to Oct. 16, 1996
Ming-Deh Huang , Dept. of Comput. Sci., Univ. of Southern California, Los Angeles, CA, USA
Yiu-Chung Wong , Dept. of Comput. Sci., Univ. of Southern California, Los Angeles, CA, USA
We consider the following polynomial congruences problem: given a prime p, and a set of polynomials f/sub 1/,...,f/sub m//spl isin/F/sub p/[x/sub 1/,...,x/sub n/] of total degree at most d, solve the system f/sub 1/=...=f/sub m/=0 for solution(s) in F/sub p//sup n/. We give a randomized algorithm for the decision version of this problem. When the system has F/sub p/-rational solutions our algorithm finds one of them as well as an approximation of the total number of such solutions. For a fixed number of variables, the algorithm runs in random polynomial time with parallel complexity poly-logarithmic in d, m and p, using a polynomial number of processors. As an essential step of the algorithm, we also formulate an algebraic homotopy method for extracting components of all dimensions of an algebraic set. The method is efficiently parallelizable.
computational complexity; polynomial congruences; randomized algorithm; decision; parallel complexity; algebraic homotopy method; algebraic set
M. Huang and Y. Wong, "Solving systems of polynomial congruences modulo a large prime," Proceedings of 37th Conference on Foundations of Computer Science(FOCS), Burlington, VT, 1996, pp. 115.