Approximation algorithms for counting problems in finite fields

Abstract

We present some recent results on the computational complexity of the generic algebraic problems of estimating the number of zeros and nonzeros of multivariate polynomials over finite fields GF[q]. We design the first polynomial time (e, 6)approximation algorithms for these two generic problems. This gives the first efficient computational method for estimating the number of points on algebraic varieties over small finite fields other than GF[2] (like GF[3]), the cases important for various approximation methods in the algebraic circuits and the coding theory. The algorithms are based on the proof of tight lower bounds for the number of nonzeros of polynomials over GF[q] in the function of the number of terms only. These lower bounds could be also of independent algebraic or geometric interest. Some further applications of our results have being also discussed.