The Logarithmic A-Discriminant

We extend the GKZ theory of discriminants to the setting of Euler integrals where both coefficients and exponents are variables. The integrand is the likelihood function of a toric model in algebraic statistics, whose number of critical points is the volume of the Newton polytope. Our logarithmic A-discriminant vanishes when two critical points come together. This discriminant is a hypersurface, parametrized by the logarithmic Horn uniformization. Hurwitz forms of toric varieties yield a degree formula, and a description of leading coefficients. We emphasize applications and computations.