Beyond the Gaussian

Abstract

In this paper we present a non-Gaussian integral based on a cubic polynomial, instead of a quadratic, and give a fundamental formula in terms of its discriminant. It gives a mathematical reinforcement to the recent result by Morozov and Shakirov. We also present some related results. This is simply one modest step to go beyond the Gaussian but it already reveals many obstacles related with the big challenge of going further beyond the Gaussian.