Inverse polynomials of symmetric numerical semigroups

Abstract: This is a joint work with Kazufumi Eto (Nippon Institute of Technology). This work was inspired by the talk of M.E. Rossi (Univ. Genova) at VCAS on Dec. 1, 2020. Let $H \subset \mathbb N$ be a numerical semigroup ring and $k[H]$ be its semigroup ring over any field $K.$ If $H = ⟨n_1, \ldots,n_e)$, we express $k[H]$ as $k[H] = k[x_1,\ldots,x_e]/I_H$ and we want to express $k[H]/(t^h)$ by ”Inverse polynomials” of Macaulay. We study the defining ideal of a numerical semigroup ring $k[H]$ using the inverse poly- nomial attached to the Artinian ring $k[H]/(t^h)$ for $h \in H_+$. I believe this method to express by inverse polynomials is very powerful and can be used for many purposes. At present, we apply this method for the following cases. (1) To give a criterion for H to be symmetric or almost symmetric. (2) Characterization of symmetric numerical semigroups of small multiplicity. (3) A new proof of Bresinsky’s Theorem for symmetric semigroups generated by 4 elements.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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