Restricted partition functions and identities for degrees of syzygies in numerical semigroups

Abstract

We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series of numerical semigroup \(\langle d_1,\ldots ,d_m\rangle \), \(m\ge 2\), generated by an arbitrary set of positive integers \(\left\{ d_1, \ldots ,d_m\right\} \), \(\gcd (d_1,\ldots ,d_m)=1\). These identities are obtained by studying the rational representation of the Hilbert series and the quasipolynomial representation of the Sylvester waves in the restricted partition function. In the cases of symmetric semigroups and complete intersections, these identities become more compact; for the latter we find a simple identity relating the degrees of syzygies with elements of generating set \(\left\{ d_1,\ldots ,d_m\right\} \) and give a new lower bound for the Frobenius number.