Numerical semigroups generated by squares, cubes and quartics of three consecutive integers

TL;DR

This paper derives polynomial formulas for key algebraic invariants of numerical semigroups generated by the squares, cubes, and quartics of three consecutive integers, providing explicit expressions for their relations, syzygies, Frobenius numbers, and genera.

Contribution

It introduces explicit polynomial representations for the minimal relations, syzygy degrees, Frobenius numbers, and genera of these specific numerical semigroups, expanding understanding of their algebraic structure.

Findings

Polynomial formulas for minimal relations of the semigroups.

Explicit expressions for degrees of syzygies in the Hilbert series.

Formulas for Frobenius numbers and genera of the semigroups.

Abstract

We derive the polynomial representations for minimal relations of generating set of numerical semigroups R_n^k=<(n-1)^k,n^k,(n+1)^k>, k=2,3,4, n>2. We find also the polynomial representations for degrees of syzygies in the Hilbert series H(z,R_n^k) of these semigroups, their Frobenius numbers F(R_n^k) and genera G(R_n^k).