Wilf's question in numerical semigroups $S_3$ revisited and inequalities for rescaled genera

TL;DR

This paper investigates properties of three-generated numerical semigroups, providing new bounds for Frobenius numbers and genera, and offers an affirmative answer to Wilf's question in this context.

Contribution

It offers a new proof of Wilf's question for $S_3$ and establishes bounds for Frobenius numbers and rescaled genera using syzygy degree identities.

Findings

Affirmative answer to Wilf's question for $S_3$

Lower bounds for Frobenius numbers in $S_3$

Bounds for rescaled genera

Abstract

We consider numerical semigroups $S_3 = \langle d_1,d_2,d_3\rangle$, minimally generated by three positive integers. We revisit the Wilf question in $S_3$ and, making use of identities for degrees of syzygies of such semigroups, give a short proof of existence of an affirmative answer. We find also the lower bound for Frobenius numbers of $S_3$ and upper and lower bounds for rescaled genera.