PreprintCC BY 4.0arXiv
Characterizing Steiner Systems via Betti Numbers of Monomial Ideals
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[x1, …, xq2+q+1] generated by squarefree monomials of degree q+1 has Betti numbers bs(S/M) = q2+q+1 choose s for all s=0,…,q then q is a prime power.
Related
BibTeX
@misc{alesandroni2026characterizing,
title = {{Characterizing Steiner Systems via Betti Numbers of Monomial Ideals}},
author = {Guillermo Alesandroni and Christopher Chin and Noah Ripke},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31783},
eprint = {2609.31783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31783},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterizing Steiner Systems via Betti Numbers of Monomial Ideals
AU - Guillermo Alesandroni
AU - Christopher Chin
AU - Noah Ripke
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.31783
UR - https://arxiv.org/abs/2609.31783
AB - 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, …, x_q^2+q+1] generated by squarefree monomials of degree q+1 has Betti numbers b_s(S/M) = q^2+q+1 choose s for all s=0,…,q then q is a prime power.
ER -