Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Factorization Statistics of Restricted Polynomial Specializations over Large Finite Fields (1810.07512v2)

Published 17 Oct 2018 in math.NT

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 $S\subset\mathbf{F}_pn$ is a subset, in the limit $p\to\infty$ and $\mathrm{deg} F$ fixed. We show that for a sufficiently large and regular subset $S\subset\mathbf{F}_pn$, 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. $S=\mathbf{F}_pn$) 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.

Summary

We haven't generated a summary for this paper yet.