Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The valuations of the near octagon I4. | The valuations of the near octagon I4 Bart De Bruyn and Pieter Vandecasteele Department of Pure Mathematics and Computer Algebra Ghent University Gent Belgium bdb@ Submitted Jun 16 2006 Accepted Aug 11 2006 Published Aug 25 2006 Mathematics Subject Classifications 51A50 51E12 05B25 Abstract The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric Q 2n 2 n 2 which are not contained in a given hyperbolic quadric Q 2n 1 2 c Q 2n 2 define a sub near polygon I of the dual polar space DQ 2n 2 . It is known that every valuation of DQ 2n 2 induces a valuation of I . In this paper we classify all valuations of the near octagon I4 and show that they are all induced by a valuation of DQ 8 2 . We use this classification to show that there exists up to isomorphism a unique isometric full embedding of I into each of the dual polar spaces DQ 2n 2 and DH 2n 1 4 . 1 Introduction Basic Definitions A near polygon is a partial linear space 5 P L I I c P X L with the property that for every point x 2 P and every line L 2 L there exists a unique point on L nearest to x. Here distances are measured in the point graph or collinearity graph r of 5. If d is the diameter of r then the near polygon is called a near 2d-gon. The unique near 0-gon consists of one point no lines . The near 2-gons are precisely the lines. Near quadrangles are usually called generalized quadrangles Payne and Thas 15 . Near polygons were introduced by Shult and Yanushka 17 because of their connection with the so-called tetrahedrally closed line systems in Euclidean spaces. A detailed treatment of the basic theory of near polygons can be found in the recent book of the author 4 . If x1 and x2 are two points of a near polygon 5 then d x1 x2 denotes the distance between x1 and x2 in the point graph . If X1 and X2 are two nonempty sets of points then d X1 X2 denotes the minimal distance between a point of X1 and a point of X2. If Postdoctoral Fellow of the Research Foundation - .
