Year: 2026
Type: preprint
Abstract: We study the structure and distribution of atoms in the monoid $\mathcal{M}{\rm on}(R)$ of nonzero monomial ideals of the polynomial ring $R=K[X_1,\ldots, X_N]$, with $N\ge 2$. We characterize the atoms with at most four minimal generators and introduce the density $d(N,μ)$ of atoms among monomial ideals with $μ$ minimal generators. For $μ\le 4$, we prove the existence of this density and determine its asymptotic behavior as $N\to\infty$. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of $\mathcal{M}{\rm on}(R)$ generated by equigenerated ideals.
Source: arXiv (Cornell University)
Authors: Bogdanovic, Nikola, Cossu, Laura
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213435573