TAILIEUCHUNG - Báo cáo toán học: "Counting points of slope varieties over finite fields"

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

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