Factorization statistics of restricted polynomial specializations over large finite fields

For a polynomial F(t, A1, …, An) ∈ F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}$$\end{document}p[t, A1, …, An] (p being a prime number) we study the factorization statistics of its specializations F(t,a1,…,an)∈Fp[t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\left({t,{a_1}, \ldots ,{a_n}} \right) \in {\mathbb{F}_p}\left[t \right]$$\end{document} with (a1, …, an) ∈ S, where S⊂Fpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subset \mathbb{F}_p^n$$\end{document} is a subset, in the limit p → ∞ and deg F fixed. We show that for a sufficiently large and regular subset S⊂Fpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \subset \mathbb{F}_p^n$$\end{document}, e.g., a product of n intervals of length H1, …, Hn with ∏i=1nHn>pn−1/2+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod\nolimits_{i = 1}^n {{H_n} > {p^{n - 1/2 +\epsilon}}} $$\end{document}, the factorization statistics is the same as for unrestricted specializations (i.e., S=Fpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S = \mathbb{F}_p^n$$\end{document}) 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.