On the Dispersions of the Polynomial Maps over Finite Fields

We investigate the distributions of the different possible values of polynomial maps F(q)(n) -> F(q), x bar right arrow P(x). In particular, we are interested in the distribution of their zeros, which are somehow dispersed over the whole domain F(q)(n). We show that if U is a "not too small" subspace of F(q)(n) (as a vector space over the prime field F(p)), then the derived maps F(q)(n)/U -> F(q), x + U bar right arrow Sigma((x) over bar is an element of x+U) P((x) over tilde) 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 is an element of F(q)(n) of systems P(1),..., P(m) is an element of F(q)[X(1),..., X(n)] of polynomials over finite fields F(q). The simultaneous zeros are distributed over all elements of certain partitions (factor spaces) F(q)(n)/U of F(q)(n). vertical bar F(q)(n)/U vertical bar is then Warning's well known lower bound for the number of these zeros.