Symmetric (not Complete Intersection) Numerical Semigroups Generated by Six Elements

TL;DR

This paper investigates symmetric numerical semigroups generated by six elements, deriving inequalities for syzygy degrees and establishing a lower bound for Frobenius numbers, especially when the Watanabe lemma applies.

Contribution

It provides new inequalities and bounds for symmetric six-generated numerical semigroups, extending understanding beyond complete intersection cases.

Findings

Derived inequalities for syzygy degrees

Established lower bounds for Frobenius numbers

Strengthened bounds under Watanabe lemma conditions

Abstract

We consider symmetric (not complete intersection) numerical semigroups S_6, generated by a set of six positive integers {d_1,...,d_6}, gcd(d_1,...,d_6)=1, and derive inequalities for degrees of syzygies of such semigroups and find the lower bound for their Frobenius numbers. We show that this bound may be strengthened if S_6 satisfies the Watanabe lemma.