Toric Ideals of Characteristic Imsets via QuasiIndependence Gluing
Abstract
Characteristic imsets are 01 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasiindependence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.01834
 arXiv:
 arXiv:2209.01834
 Bibcode:
 2022arXiv220901834H
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 62R01;
 13P10;
 62A09;
 13P25
 EPrint:
 19 pages, 7 figures