On the Saito Number of Plane Curves
Yohann Genzmer, Marcelo Escudeiro Hernandes · Bulletin of the Brazilian Mathematical Society New Series · 2025
Abstract In this study, we examine the Saito number of a plane curve and propose a method for determining the minimum Saito number within a given equisingularity class, resulting in an actual algorithm. In certain cases, we also offer explicit formulas for this number. In addition, if $$ u _0$$ ν 0 and $$ u _1$$ ν 1 are coprime integers with $$1\le u _00$$ N > 0 , we show that there exists a plane curve equisingular to the curve $$\begin{aligned} y^{N u _0}-x^{N u _1}=0 \end{aligned}$$ y N ν 0 - x N ν 1 = 0 such that its Saito number is equal to k, for any $$1\le k\le \left[ \frac{N u _0}{2}\right] .$$ 1 ≤ k ≤ N ν 0 2 .