Betti numbers for numerical semigroup rings

TL;DR

This paper surveys the known results concerning the size of Betti numbers in numerical semigroup rings and their tangent cones, highlighting key findings and open questions in the area.

Contribution

It provides a comprehensive overview of existing research on Betti numbers in numerical semigroup rings and tangent cones, summarizing current knowledge and identifying gaps.

Findings

Betti numbers vary significantly across different numerical semigroup rings.

Tangent cones often exhibit complex Betti number patterns.

Open problems remain regarding bounds and exact values of Betti numbers.

Abstract

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.