On the Distribution of Values and Zeros of Polynomial Systems over Arbitrary Sets

الأصل · EN

Let G₁,..., Gₙ ∈ [X₁,...,Xₘ] be n polynomials in m variables over the finite field 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 (p)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.