Symmetric (not Complete Intersection) Semigroups Generated by Five Elements

TL;DR

This paper investigates symmetric numerical semigroups generated by five elements, deriving inequalities for syzygy degrees and establishing bounds for Frobenius numbers, with a focus on a special case satisfying the Watanabe Lemma.

Contribution

It introduces new inequalities for syzygy degrees and provides improved lower bounds for Frobenius numbers in specific symmetric semigroups.

Findings

Derived inequalities for degrees of syzygies in S_5

Established a lower bound F_5 for Frobenius numbers

Showed the bound F_{5w} for W_5 is stronger than F_5

Abstract

We consider symmetric (not complete intersection) numerical semigroups S_5, generated by five elements, and derive inequalities for degrees of syzygies of S_5 and find the lower bound F_5 for their Frobenius numbers. We study a special case W_5 of such semigroups, which satisfy the Watanabe Lemma, and show that the lower bound F_{5w} for the Frobenius number of the semigroup W_5 is stronger than F_5.