# Ralf Fröberg: Curves (mostly plane) and semigroups

**Time: **
Mon 2023-02-06 15.00 - 16.00

**Location: **
Zoom

**Video link: **
Meeting ID: 667 9231 0361

**Participating: **
Ralf Fröberg, Stockholm University

**Abstract.**

A numerical semigroup *S* is a subset of \(\mathbb N\) such that if \(m,n\in S \text{ then } m+n\in S\), \(0\in S\), and \({\mathbb N}\setminus S\) is finite. It is easy to see that such a semigroup $S$ has a unique minimal system of generators \(\{ n_1,\ldots,n_r\}\), such that *S* consists of all linear combinations of the generators with nonnegative integer coefficients. If \(S=\langle n_1,\ldots,n_r\rangle\), the semigroup ring \(k[[S]]\) is then \(k[[t^{n_1},\ldots,t^{n_r}]]\). Let *f(x,y)* be an irreducible power series in \(k[[x,y]]\) (or analytic function around the origin) which has a singularity in $(0,0)$. For such an *f*one can define a numerical semigroup which determines the topological type of the singularity. (The semigroup in fact consists of all intersection numbers *I(f,h)*, where *h* is prime to *f*.) Also for parametrized curves in higher dimensional spaces one can define a numerical semigroup, which sometimes says a lot about the curve. I will describe this connection in the first part of my talk. Then I will mention some results and open questions about numerical semigroups and semigroup rings.