On the dispersions of the polynomial maps over finite fields

We investigate the distributions of the different possible values of polynomial maps F_qⁿ → F_q, x → P(x). In particular, we are interested in the distribution of their zeros, which are somehow dispersed over the whole domain F_qn. We show that if U is a "not too small" subspace of F_qⁿ (as a vector space over the prime field F_p), then the derived maps F _qⁿ/U → F_q , x + U → ∑_x̃∈x+UP(x∼) 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 ∈ F_qⁿ of systems P₁,..., P_m ∈ F_q[X₁,..., X_n] of polynomials over finite fields F_q. The simultaneous zeros are distributed over all elements of certain partitions (factor spaces) F_qⁿ/U of F^q_n . |F_qⁿ/U| is then Warning's well known lower bound for the number of these zeros.