On the distribution of values and zeros of polynomial systems over arbitrary sets

Bryce Kerr, Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Let G1,...,Gn∈Fp[X1,...,Xm] be n polynomials in m variables over the finite field Fp of p elements. A result of É. Fouvry and N.M. Katz shows that under some natural condition, for any fixed ε and sufficiently large prime p the vectors of fractional parts({G1(x)p},...,{Gn(x)p}),x∈Γ, are uniformly distributed in the unit cube [0,1]n for any cube Γ ∈ [0,p -1]m with the side length h ≥ p1/2(logp)1 +ε. Here we use this result to show the above vectors remain uniformly distributed, when x runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.

Original languageEnglish
Pages (from-to)2863-2873
Number of pages11
JournalJournal of Number Theory
Volume133
Issue number9
DOIs
Publication statusPublished - Sep 2013

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