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 $(-2)^r\chi_M( \textstyle\frac{1}{2})$, 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.