The generic value set of a two-generator semigroup

TL;DR

This paper introduces a recursive method to explicitly compute the minimal generators of the generic value set for plane curve germs with two-generator semigroups, providing formulas for key invariants and comparing computational approaches.

Contribution

It presents a new recursive algorithm for calculating minimal generators of the generic value set in two-generator semigroups, with explicit formulas and comparisons to existing methods.

Findings

Explicit formulas for minimal generators and conductor.

Recursive algorithm for generic Tjurina number calculation.

Comparison with existing computational methods.

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.