Factorization Statistics of Restricted Polynomial Specializations over Large Finite Fields

Abstract

For a polynomial $F(t,A_1,\dots ,A_n)\in {\mathbf{F}}_p[t,A_1,\dots ,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\dots ,a_n)\in {\mathbf{F}}_p[t]$$ with $(a_1,\dots ,a_n)\in S$, where $S\subset {\mathbf{F}}_p^n$ is a subset, in the limit $p\to \infty$ and $\mathrm{deg}F$ fixed. We show that for a sufficiently large and regular subset $S\subset {\mathbf{F}}_p^n$, e.g. a product of $n$ intervals of length $H_1,\dots ,H_n$ with ${\prod }_{i=1}^n{H}_n>p^{n-1/2+\epsilon }$, the factorization statistics is the same as for unrestricted specializations (i.e. $S={\mathbf{F}}_p^n$) 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.