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 $1\le k\le \left[\frac{N{\nu }_0}{2}\right]$ 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 $k$.