TY - JOUR

T1 - On the distribution of values and zeros of polynomial systems over arbitrary sets

AU - Kerr, Bryce

AU - Shparlinski, Igor E.

PY - 2013/9

Y1 - 2013/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84876722743&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2013.02.012

DO - 10.1016/j.jnt.2013.02.012

M3 - Article

VL - 133

SP - 2863

EP - 2873

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 9

ER -