Exponential Varieties

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We develop a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials and their integral representations on the hyperbolicity cone. We compare the multidegrees and ML degrees of the gradient map for such polynomials.