On the number of regions and multiplicities of vertices in plane arrangements

TL;DR

This paper introduces a new linear inequality related to the vertices of pseudoline arrangements in the projective plane, providing an elementary proof and an algorithm for lower bounds on the number of regions, with applications to Martinov's theorem.

Contribution

It presents a novel linear inequality for pseudoline arrangements, an elementary proof, and an algorithm for deriving lower bounds on the number of regions.

Findings

New linear inequality for pseudoline arrangements

Elementary proof of the inequality

Algorithm for lower bounds on regions

Abstract

For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We present an algorithm for producing lower bounds of the number of regions basing on linear on $t_i$ inequalities like the above-mentioned. Lower bounds arise in connection with Martinov theorem on the set of all possible numbers of regions and we show how the new bounds may be applied in it.