Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 022, 12 pages      arXiv:0912.2135

Beyond the Gaussian

Kazuyuki Fujii
Department of Mathematical Sciences, Yokohama City University, Yokohama, 236-0027 Japan

Received January 12, 2011, in final form February 28, 2011; Published online March 04, 2011

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.

Key words: non-Gaussian integral; renormalized integral; discriminant; cubic equation.

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