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

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.