### On the Dispersions of the Polynomial Maps over Finite Fields

#### Abstract

We investigate the distributions of the different possible values of polynomial maps ${\Bbb F}_q^n\longrightarrow{\Bbb F}_q$, $x\longmapsto P(x)$. In particular, we are interested in the distribution of their zeros, which are somehow dispersed over the whole domain ${\Bbb F}_q^n$. We show that if $U$ is a "not too small" subspace of ${\Bbb F}_q^n$ (as a vector space over the prime field ${\Bbb F}_p$), then the derived maps ${\Bbb F}_q^n/U\longrightarrow{\Bbb F}_q$, $x+U\longmapsto\sum_{\tilde x\in x+U}P(\tilde x)$ 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 $x\in{\Bbb F}_q^n$ of systems $P_1,\dots,P_m\in{\Bbb F}_q[X_1,\dots,X_n]$ of polynomials over finite fields ${\Bbb F}_q$. The simultaneous zeros are distributed over all elements of certain partitions (factor spaces) ${\Bbb F}_q^n/U$ of ${\Bbb F}_q^n$. $|\,{\Bbb F}_q^n/U|$ is then Warning's well known lower bound for the number of these zeros.