Ordinarization numbers of numerical semigroups

Abstract

There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amorós has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amorós for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval.

Figures (1)

• Figure 1: ${\mathcal{T}}_7$, the ordinarization tree of numerical semigroups of genus $7$. This figure was created using the website of Bras-Amorós for drawing trees of numerical semigroups BA_draw.

Theorems & Definitions (60)

• Theorem 1.1
• Conjecture 1.2: Bras-Amorós
• Conjecture 1.3: Bras-Amorós
• Proposition 1.4
• Conjecture 1.5: Bras-Amoros
• Proposition 1.6
• Proposition 2.1
• Theorem 2.3
• Theorem 2.5
• Lemma 2.6