Reciprocal maximum likelihood degrees of diagonal linear concentration models

Abstract

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model $\mathcal{L}\subseteq {\mathbb{C}}^n$ of dimension $r$ is equal to $(-2{)}^r{\chi }_M(\frac{1}{2})$, where ${\chi }_M$ is the characteristic polynomial of the matroid $M$ associated to $\mathcal{L}$. In particular, this establishes the polynomiality of the rmld for general diagonal linear concentration models, positively answering a question of Sturmfels, Timme, and Zwiernik.

13 pages, comments welcome To appear in: Le Matematiche, vol. 76 (2) (special issue on Linear Spaces of Symmetric Matrices)