Beyond the Gaussian

TL;DR

This paper introduces a non-Gaussian integral based on cubic polynomials, providing a fundamental formula involving the discriminant, and discusses the challenges of extending beyond Gaussian models.

Contribution

It presents a novel cubic polynomial-based integral and a fundamental formula, advancing the mathematical understanding beyond Gaussian assumptions.

Findings

Derived a fundamental formula in terms of the discriminant for the cubic-based integral

Reinforced recent results by Morozov and Shakirov

Highlighted obstacles in extending beyond Gaussian models

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.