Factorization statistics of restricted polynomial specializations over large finite fields

For a polynomial F(t, A1, …, An) ∈ $$\mathbb{F}$$ p[t, An] (p being prime number) we study the factorization statistics of its specializations $$F\left({t,{a_1}, \ldots ,{a_n}} \right) \in {\mathbb{F}_p}\left[t \right]$$ with (a1, an) S, where $$S \subset \mathbb{F}_p^n$$ is subset, in limit p → ∞ and deg F fixed. We show that for sufficiently large regular subset , e.g., product n intervals length H1, Hn $$\prod\nolimits_{i = 1}^n {{H_n} > {p^{n - 1/2 +\epsilon}}} $$ same as unrestricted (i.e., ) up to small error. This generalization well-known Pólya-Vinogradov estimate number quadratic residues modulo an interval.