Year: 2026
Type: preprint
Abstract: Given a point set $[n]$ and a collection $B$ of $k$-subsets of $[n]$, we create a monomial ideal whose Betti numbers completely determine whether the pair $([n],B)$ is a Steiner system $S(t,k,n)$. In particular, our construction characterizes Steiner triple systems and Steiner quadruple systems solely on the basis of Betti numbers of monomial ideals. Moreover, these monomial ideals allow us to reformulate the renowned prime power conjecture as follows: If a monomial ideal $M$ of $S = K[x_1, \dots, x_{q^2+q+1}]$ generated by squarefree monomials of degree $q+1$ has Betti numbers $\mathrm{b}_s(S/M) = {{q^2+q+1} \choose s} \text{ for all } s=0,\ldots,q$ then $q$ is a prime power.
Source: arXiv (Cornell University)
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214769900