On the dispersions of the polynomial maps over finite fields

Uwe Schauz

The electronic journal of combinatorics, Tome 15 (2008)

Résumé

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.

Détail

DOI : 10.37236/869

Classification : 13M10, 11G25, 11D79, 15A15