The generic value set of a two-generator semigroup

Abstract

We give a simple recursion for the minimal generators $G=\{g_i\}$ of the generic value set $\Lambda_{gen}$ of a plane curve germ $C$ with a two-generator semigroup $\Gamma \langle p, m \rangle$. The main result provides for explicit calculation of all $g_i$ and shows that they are in fact minimal generators. Explicit formulas for the cardinality of the minimal generators $|G|$ and for the conductor $c(\Lambda_{gen})$ of $\Lambda_{gen}$ are given. The recursion can be used to compute the generic Tjurina number $\tau_{gen}$ in this case, a method we compare to that given by Alberich-Carrami\~nana, Almir\'on, Blanco, and Melle-Hern\'andez (arXiv:1904.02652). The main result is based on the non-explicit algorithm and ideas of Delorme.