Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Counting points of slope varieties over finite fields. | Counting points of slope varieties over finite fields Thomas Enkosky Department of Mathematics University of Kansas Lawrence KS . tenkosky@ Submitted Oct 10 2010 Accepted Dec 7 2010 Published Jan 5 2011 Mathematics Subject Classification 05C30 14G15 05A19 Abstract The slope variety of a graph is an algebraic set whose points correspond to drawings of that graph. A complement-reducible graph or cograph is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on n vertices over F2 and the complement-reducible graphs on n vertices. 1 Introduction Fix a field F and a positive integer n. Let P1 x1 y1 . Pn xn yn be points in the plane F2 such that the xi are distinct. Let L12 . Ln-1 n be the n lines in F2 where Lj is the line through Pi and Pj. The slope variety SF Kn is the set of possible n -tuples mi 2 . mn-1 n where mi j Xi-x. denotes the slope of Li j if F is a finite field with q elements then we use the notation Sq Kn . Over an algebraically closed field the slope variety is the set of simultaneous solutions of certain polynomials TW called tree polynomials 4 5 indexed by wheel subgraphs of the complete graph Kn. A k-wheel is a graph formed from a cycle of length k by introducing a new vertex adjacent to all vertices in the cycle. The ideal generated by all tree polynomials is radical 5 Theorem . It is conjectured and has been verified experimentally for n 9 that the ideal of all tree polynomials is in fact generated by the subset tq 5 where Q is a 3-wheel equivalently a 4-clique in Kn . The tree polynomials have integer coefficients which raises the question of counting their solutions over a finite field. Let Fq be the field with q elements. In this article we count the solutions of the tree polynomials over F2 and give some generalizations for q 2. 2 When the slope variety is considered over Fq the points correspond to drawings in Fq whose slopes are in Fq . These