Factorization Statistics of Restricted Polynomial Specializations over Large Finite Fields

TL;DR

This paper investigates how the factorization patterns of polynomial specializations over large finite fields behave when restricted to large subsets, showing they match the unrestricted case under certain size conditions.

Contribution

It generalizes the Pólya-Vinogradov estimate to polynomial specializations over finite fields with restricted variable sets, establishing conditions for statistical similarity.

Findings

Factorization statistics match unrestricted case for large regular subsets

Results hold as prime p tends to infinity with fixed polynomial degree

Generalizes classical quadratic residue estimates to polynomial specializations

Abstract

For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with $(a_1,\ldots,a_n)\in S$, 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 $H_1,\ldots,H_n$ with $\prod_{i=1}^nH_n>p^{n-1/2+\epsilon}$, 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 P\'olya-Vinogradov estimate of the number of quadratic residues modulo $p$ in an interval.