On the dispersions of the polynomial maps over finite fields (Q1010882)

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