Solving systems of polynomial congruences modulo a large prime
Conference
·
OSTI ID:457643
- Univ. of Southern California, Los Angeles, CA (United States)
We consider the following polynomial congruences problem: given a prime p, and a set of polynomials f{sub 1}, . , . , f{sub m} {element_of} 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{sup n}{sub p}. 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.
- OSTI ID:
- 457643
- Report Number(s):
- CONF-961004--; CNN: Grant CCR-9412383
- Country of Publication:
- United States
- Language:
- English
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