DIMACS Theoretical Computer Science Seminar Title: Solvability of polynomial equations over finite fields Speaker: Neeraj Kayal, IAS Date: Wednesday, October 4, 2006 11:00am - 12:00pm Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ We investigate the complexity of the following polynomial polynomial solvability problem - given a finite field and a set of polynomials of total degree at most an , determine the existence of -solution to the system of equations That is determine if there exists a point such that The problem is easily seen to be NP-complete even when the field size is and the degree of each polynomial is also bounded by . Here we investigate the deterministic complexity of this problem when the number of variables in the input is bounded. We show that for any fixed , there is a deterministic algorithm for this problem whose running time is bounded by a polynomial in . Moreover the algorithm can be implemented parallely to get a family of P-uniform circuits of size and depth for this problem

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# Solvability of polynomial equations over finite fields