Equations and multidegrees for inverse symmetric matrix pairs

TL;DR

This paper calculates the equations and multidegrees of the variety parametrizing inverse symmetric matrix pairs, providing new insights and an alternative proof for a conjecture on maximum likelihood degree polynomiality.

Contribution

It introduces explicit equations and multidegrees for the inverse symmetric matrix pairs variety and offers an alternative proof for a key conjecture in algebraic statistics.

Findings

Derived equations and multidegrees for the variety

Provided an alternative proof for the polynomiality of maximum likelihood degree

Confirmed the conjecture of Sturmfels and Uhler

Abstract

We compute the equations and multidegrees of the biprojective variety that parametrizes pairs of symmetric matrices that are inverse to each other. As a consequence of our work, we provide an alternative proof for a result of Manivel, Michalek, Monin, Seynnaeve and Vodi\v{c}ka that settles a previous conjecture of Sturmfels and Uhler regarding the polynomiality of maximum likelihood degree.