Frobenius numbers and the first Hilbert coefficients of certain numerical semigroup rings

TL;DR

This paper derives explicit formulas for the Frobenius number and the first Hilbert coefficient of certain numerical semigroup rings generated by three elements, enhancing understanding of their algebraic properties.

Contribution

It provides new explicit formulas for Frobenius numbers and Hilbert coefficients for a class of numerical semigroup rings, under specific conditions.

Findings

Explicit Frobenius number formulas for semigroups generated by three elements.

Explicit formulas for the first Hilbert coefficient of the associated rings.

Enhanced understanding of algebraic invariants of numerical semigroup rings.

Abstract

Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=\left<a,a+1,b\right>$ and and the associated numerical semigroup ring $R=k[[H]]$. Under the certain conditions, we provide explicit formulas for the Frobenius number of $H$ and for the first Hilbert coefficient of $R$.