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arXiv · 2601.15128

Decomposing Determinantal Varieties from Statistics via Matroid Theory

Per Alexandersson·Yulia Alexandr·Emiliano Liwski·Fatemeh Mohammadi·Pardis Semnani
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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.

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Per Alexandersson, Yulia Alexandr, Emiliano Liwski, Fatemeh Mohammadi, Pardis Semnani. 2026-01-21. Decomposing Determinantal Varieties from Statistics via Matroid Theory. https://arxiv.org/abs/2601.15128

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