Factorization statistics of restricted polynomial specializations over large finite fields

For a polynomial F(t, A₁, …, A_n) ∈ F_p[t, A₁, …, A_n] (p being a prime number) we study the factorization statistics of its specializations F(t, a₁, … , a_n) ∈ F_p[t] with (a₁, …, a_n) ∈ S, where S⊂Fpn is a subset, in the limit p → ∞ and deg F fixed. We show that for a sufficiently large and regular subset S⊂Fpn, e.g., a product of n intervals of length H₁, …, H_n with ∏i=1nHn>pn−1/2+ϵ, the factorization statistics is the same as for unrestricted specializations (i.e., S=Fpn) up to a small error. This is a generalization of the well-known Pólya-Vinogradov estimate of the number of quadratic residues modulo p in an interval.