Year: 2025
Type: article
Abstract: Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups of genus $g$, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\]for some constant $C_d > 0$, which can be made explicit. To do this, we extend the notion of multiplicity and depth to generalized numerical semigroups and show our lower bound is sharp for semigroups of depth 2. We also show other bounds on special classes of semigroups by introducing partition labelings, which extend the notion of Kunz words to the general setting.
DOI: 10.37236/12287
Author: Sean Li
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Awards: G319878833, G7845798442
OpenAlex ID: W4413247359