Inverse polynomials of numerical semigroup rings

TL;DR

This paper introduces inverse polynomials for numerical semigroup rings, providing new methods to analyze their defining ideals, Gorenstein properties, and symmetry, with applications including a new proof of Bresinsky's theorem.

Contribution

It defines inverse polynomials for semigroup rings and uses them to characterize Gorenstein, almost Gorenstein, and symmetric semigroups, offering novel insights and proofs.

Findings

Characterization of defining ideals via inverse polynomials.

Criteria for almost Gorenstein and symmetric semigroups.

New proof of Bresinsky's theorem for codimension 3 Gorenstein rings.

Abstract

Let H = <n_1,...,n_e> be a numerical semigroup generated by e elements. Let k[H]= k[x_1, .... , x_e]/I_H = S/I_H be the semigroup ring of H over k. We define inverse polynomial J_{H,h} for h in H and express the defining ideal of I_H using Ann_S (J_{H,h}). In particular, if k[H] is Gorenstein the defining ideal of I_H + (t^h) is Ann_S (J_{H, F(H)+h}), where F(H) is the Frobenius number of H ( = a(k[H]), the a -invariant of k[H]). We apply this to (1) evaluate number of generators of I_H, (2) characterize if k[H] is almost Gorenstein (H is almost symmetric), (3) characterize symmetric semigroups of small multiplicity. Also We give a new proof of Bresinsky's Theorem on Gorenstein semigroup rings of codimension 3 using inverse polynpmial.