Decomposing Determinantal Varieties from Statistics via Matroid Theory
Combinatorics
2026-01-22 v1 Commutative Algebra
Algebraic Geometry
Statistics Theory
Statistics Theory
Abstract
We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gr\"obner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.
Cite
@article{arxiv.2601.15128,
title = {Decomposing Determinantal Varieties from Statistics via Matroid Theory},
author = {Per Alexandersson and Yulia Alexandr and Emiliano Liwski and Fatemeh Mohammadi and Pardis Semnani},
journal= {arXiv preprint arXiv:2601.15128},
year = {2026}
}