Algebraic Stories — Hilbert series of numerical semigroup rings

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb N_0=\{0,1,\ldots\}$, and let $\mathcal S\subset\mathbb N_0$ be an additive submonoid containing zero with finite complement. With minimal generators $s_1,\ldots,s_e$, its semigroup ring over a field $K$ is $K[\mathcal S]=K[t^{s_1},\ldots,t^{s_e}]$. Set \[P_{\mathcal S}(z)=(1-z)\sum_{s\in\mathcal S}z^s. \] The finite complement makes $P_{\mathcal S}$ a polynomial. Call $\mathcal S$ cyclotomic if this polynomial factors over $\mathbb Q$ into cyclotomic polynomials (equivalently, all its complex roots lie on the unit circle). The ring is a complete intersection if the kernel of $K[y_1,\ldots,y_e]\to K[t]$, $y_i\mapsto t^{s_i}$, is generated by a regular sequence of length $e-1$.

Statement recorded in the supplied source.

Is $\mathcal S$ cyclotomic if and only if $K[\mathcal S]$ is a complete intersection? [S2]

Short English statement

Are cyclotomic numerical semigroups exactly those whose defining semigroup rings are complete intersections?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500017 (AMR-014-0017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. Fröberg, S. Lundqvist, A. Oneto and B. Shapiro, Algebraic Stories from One and from the Other Pockets, arXiv:1801.01692 (2018), §4.1, Conjecture 4.1. https://arxiv.org/abs/1801.01692.
• S3 E.-A. Ciolan, P. A. García-Sánchez and P. Moree, Cyclotomic numerical semigroups, arXiv:1409.5614v3 (2016), §1, Definition 1 and Conjecture 1. https://arxiv.org/abs/1409.5614.