Counting Homogeneous Einstein Metrics

We present an explicit upper bound on the number of isolated homogeneous Einstein metrics on compact homogeneous spaces whose isotropy representations consist of pairwise inequivalent irreducibles. This is the BKK bound of the corresponding system of Laurent polynomials and is found combinatorially by computing the volume of a polytope. Inspired by a connection with algebraic statistics, we describe this system's BKK discriminant in terms of the principal $A$-determinant of scalar curvature. As a consequence, we confirm the Finiteness Conjecture of B\"ohm--Wang--Ziller in special cases. In particular, we give a unified proof that it holds on all generalized Wallach spaces. Finally, using numerical algebraic geometry, we compute $G$-invariant Einstein metrics on low-dimensional full flag manifolds $G/T$, where $G$ is a compact simple Lie group and $T$ is a maximal torus.