Gaussian-like fixed point and variational properties of integral discriminants

We consider partition functions Z(g) = exp (-g(x))dx where g is a nonnegative polynomial action (a degree-2n form) vanishing only at the origin. Such integrals, known as integral discriminants, appear in statistical mechanics, quantum field theory, and the theory of exponential families. We show that the associated Boltzmann measure d$\mu$ = exp(-g(x))dx satisfies a fixed-point property identity relating in a simple manner its degree-2n moments to the coefficients of g. This generalizes familiar identities for the exponential distribution (degree-1) on the positive orthant and the Gaussian measure (degree-2). We further show that g is characterized by three variational principles, including a maximum-entropy principle under scaled moments constraints, extending the Gaussian extremality principle to arbitrary even-degree homogeneous actions. Exploiting these identities in a truncatedmoment numerical scheme (known as the Moment-SOS hierarchy), strengthens the standard semidefinite relaxations, and results in a much faster convergence, thus allowing more efficient approximations of the partition function Z(g) as well as moments of $\mu$.