FACTORIZATION STATISTICS OF RESTRICTED POLYNOMIAL SPECIALIZATIONS OVER LARGE FINITE FIELDS

For a polynomial F(t, A(1),..., A(n)) is an element of F-p[t, A(1),..., A(n)] (p being a prime number) we study the factorization statistics of its specializations F(t, a(1), ..., a(n)) is an element of F-p[t] with (a(1),..., a(n)) is an element of S, where S subset of F-p(n) is a subset, in the limit p -> infinity and deg F fixed. We show that for a sufficiently large and regular subset S subset of F-p(n), e.g., a product of n intervals of length H-1,..., H-n with Pi(n)(i=1) H-n > p(n-1/2+epsilon,) the factorization statistics is the same as for unrestricted specializations (i. e., S = F(p)(n)p) up to a small error. This is a generalization of the well-known Polya-Vinogradov estimate of the number of quadratic residues modulo p in an interval.