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{\bf Uwe Schauz}
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{\bf On the Dispersions of the Polynomial Maps over Finite Fields}
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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.



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