Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the Dispersions of the Polynomial Maps over Finite Fields. | On the Dispersions of the Polynomial Maps over Finite Fields Uwe Schauz Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia Submitted Dec 19 2007 Accepted Nov 24 2008 Published Nov 30 2008 Mathematics Subject Classifications 13M10 11G25 11D79 15A15 Abstract We investigate the distributions of the different possible values of polynomial maps Fj Fq x I P x . In particular we are interested in the distribution of their zeros which are somehow dispersed over the whole domain Fj We show that if U is a not too small subspace of Fj as a vector space over the prime field Fp then the derived maps Fq U Fq x U I Px2x 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 2 Fj of systems Pl . Pm 2 Fq X1 . Xn of polynomials over finite fields Fq . The simultaneous zeros are distributed over all elements of certain partitions factor spaces Fj U of Fj . I Fj UI is then Warning s well known lower bound for the number of these zeros. Introduction As described in the abstract we will investigate the distributions of the different possible values of polynomial maps F Fj x I P x . In particular we are interested in the distribution of their zeros in the domain Fj . It turns out that they are somehow dispersed over the whole domain Fj a property that strongly relies on the finiteness of the ground field Fj . The original goal behind this was to present a new sharpening supplementation of the following classical result due to Chevalley and Warning about the set of simultaneous zeros V x e F I P1 x Pm x 0 g of polynomials V P1 . Pm e Fj X1 . X over finite fields Fj of characteristic p m n Fq p THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R145 1 Theorem . If X 1 deg Pị n then p divides V and hence the Pi do not have one unique common zero . V 1 . This theorem goes back to a conjecture of .
