Numerical Semigroups Generated by Squares and Cubes of Three Consecutive Integers

Abstract

We derive the polynomial representations for minimal relations of the generating set of numerical semigroups \(R_n^k=\langle (n-1)^k,n^k,(n+1)^k\rangle \), \(k=2,3\). We find also the polynomial representations for degrees of syzygies in the Hilbert series \(H\left( z,R_n^k\right) \) of these semigroups, their Frobenius numbers \(F\left( R_n^k\right) \) and genera \(G\left( R_n^k\right) \). We discuss an extension of polynomial representations for minimal relations on numerical semigroups \(R_n^k\), \(k\ge 4\).