On the Saito number of plane curves

Abstract

In this work we study the \emph{Saito number} of a plane curve and we present a method to determine the minimal Saito number for plane curves in a given equisingularity class, that gives rise to an actual algorithm. In particular situations, we also provide various formulas for this number. In addition, if $\nu_0$ and $\nu_1$ are two coprime positive integers and $N>0$ then we show that for any $1\leq k\leq \left [\frac{N\nu_0}{2}\right ]$ there exits a plane curve equisingular to the curve $$y^{N\nu_0}-x^{N\nu_1}=0$$ such that its Saito number is precisely $k$.