21-25 August 2012
Portorož, Slovenia
UTC timezone
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REALIZABILITY OF (12_4, 16_3) CONFIGURATIONS

Presented by Mr. Marko BOBEN
Type: Oral presentation

Content

A \emph{combinatorial configuration} $(v_r, b_k)$ is an incidence structure with $v$ points and $b$ blocks (called lines) such that each point is incident with $r$ lines, each line is incident with $k$ points and two distinct lines are incident with at most one common point. When $v = b$ (and consequently $r = k$) we speak of \emph{symmetric} or \emph{balanced} $(v_r)$ configurations. One of the oldest problems studied in this field is possibility of a \emph{realization} of points and lines of a combinatorial configuration with points and lines in the Euclidean plane. Most of the research was done on $(v_3)$ configurations. In this case we can reduce the problem of realization to looking for (non-trivial) solutions of a single polynomial equation. In this talk we focus on $(12_4, 16_3)$ configurations. This is, after the single $(9_4, 12_3)$ configuration (affine plane of order three), the smallest possible case for $(v_4, b_3)$ configurations. Although such configurations were first studied more than 150 years ago, there are still many unresolved questions, in particular, the problem of realizability remains almost untouched. We show that the problem of realizability of $(12_4, 16_3)$ configurations involves solving a system of three polynomial equations. Each of $574$ $(12_4, 16_3)$ configurations is considered by solving the corresponding system of equations by a polynomial homotopy, using the PHCpack solver. An analysis of the results will be presented.

Place

Location: Portorož, Slovenia
Address: University of Primorska, Faculty of Tourism Studies, Obala 11a, SI-6320 Portorož - Portorose, Slovenia
Room: VP1

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