## Abstract

We investigate the distributions of the different possible values of polynomial maps F_{q}^{n} → F_{q}, x → P(x). In particular, we are interested in the distribution of their zeros, which are somehow dispersed over the whole domain F_{qn}. 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 → ∑_{x̃∈x+U}P(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 ∈ F_{q}^{n} of systems P_{1},..., P_{m} ∈ 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} . |F_{q}^{n}/U| is then Warning's well known lower bound for the number of these zeros.

Original language | English |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Electronic Journal of Combinatorics |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - 30 Nov 2008 |

Externally published | Yes |