On the Dispersions of the Polynomial Maps over Finite Fields

We investigate the distributions of the different possible values of polynomial maps Fq n −→ Fq , x 7−→ P (x) . In particular, we are interested in the distribution of their zeros, which are somehow dispersed over the whole domain Fq n . We show that if U is a “not too small” subspace of Fq n (as a vector space over the prime field Fp ), then the derived maps Fq /U −→ Fq , x + U 7−→ ∑ x̃∈x+U P (x̃) are constant and, in certain cases, not zero. Such observations lead to a refinement of Warning’s classical result about the number of simultaneous zeros x ∈ Fq n of systems P1, . . . , Pm ∈ Fq[X1, . . . , Xn] of polynomials over finite fields Fq . The simultaneous zeros are distributed over all elements of certain partitions (factor spaces) Fq /U of Fq n . |Fq /U | is then Warning’s well known lower bound for the number of these zeros. Introduction As described in the abstract, we will investigate the distributions of the different possible values of polynomial maps Fq n −→ Fq , x 7−→ P (x) . In particular, we are interested in the distribution of their zeros in the domain Fq . It turns out that they are somehow dispersed over the whole domain Fq , a property that strongly relies on the finiteness of the ground field Fq . The original goal behind this was to present a new sharpening (supplementation) of the following classical result, due to Chevalley and Warning, about the set of simultaneous zeros V := { x ∈ Fq n P1(x) = · · · = Pm(x) = 0 } of polynomials V P1, . . . , Pm ∈ Fq[X1, . . . , Xn] over finite fields Fq of characteristic p : m, n Fq , p the electronic journal of combinatorics 15 (2008), #R145 1 Theorem 0.1. If ∑m i=1 deg(Pi) < n , then p divides |V| and hence the Pi do not have one unique common zero, i.e., |V| 6= 1 . This theorem goes back to a conjecture of Dickson and Artin [Ar] and has a short and elegant proof [Scha, Theorem4.3], [Schm]. There are a lot of different sharpenings and supplementations, which follow two main streams. The first one [MSCK, MoMo, Wan, Ax, Ka] tries to improve the divisibility property and led, e.g., to the following improvement by Katz (see [MSCK, Wan, Wan2, AdSp, AdSp2, Sp] for generalizations to exponential sums): Theorem 0.2. If Σ := ∑m i=1 deg(Pi) < n and M := max 1≤i≤m (deg(Pi)) , then Σ, M