Factorization statistics of restricted polynomial specializations over large finite fields

Alexei Entin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For a polynomial F(t, A1, …, An) ∈ Fp[t, A1, …, An] (p being a prime number) we study the factorization statistics of its specializations F(t, a1, … , an) ∈ Fp[t] with (a1, …, an) ∈ S, where S⊂Fpn is a subset, in the limit p → ∞ and deg F fixed. We show that for a sufficiently large and regular subset S⊂Fpn, e.g., a product of n intervals of length H1, …, Hn with ∏i=1nHn>pn−1/2+ϵ, the factorization statistics is the same as for unrestricted specializations (i.e., S=Fpn) 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.

Original languageEnglish
Pages (from-to)37-53
Number of pages17
JournalIsrael Journal of Mathematics
Volume242
Issue number1
DOIs
StatePublished - Apr 2021

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