TY - JOUR

T1 - Factorization statistics of restricted polynomial specializations over large finite fields

AU - Entin, Alexei

N1 - Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.

PY - 2021/4

Y1 - 2021/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85101115842&partnerID=8YFLogxK

U2 - 10.1007/s11856-021-2101-9

DO - 10.1007/s11856-021-2101-9

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AN - SCOPUS:85101115842

SN - 0021-2172

VL - 242

SP - 37

EP - 53

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -