On the Saito number of plane curves

TL;DR

This paper investigates the Saito number of plane curves, introducing an algorithm to find the minimal Saito number within an equisingularity class and providing formulas for specific cases.

Contribution

It presents a new method and algorithm to determine the minimal Saito number for plane curves in a given equisingularity class, along with explicit formulas for certain cases.

Findings

An algorithm to compute minimal Saito numbers for plane curves.

Formulas for Saito numbers in particular situations.

Existence of curves with prescribed Saito numbers within a class.

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 $k$.