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


SIGMA 7 (2011), 022, 12 pages      arXiv:0912.2135      http://dx.doi.org/10.3842/SIGMA.2011.022

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

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.

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

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References

  1. Fujii K., Beyond Gaussian: a comment, arXiv:0905.1363.
  2. Morozov A., Shakirov Sh., Introduction to integral discriminants, J. High Energy Phys. 2009 (2009), no. 12, 002, 39 pages, arXiv:0903.2595.
  3. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge University Press, Cambridge, 1996.
  4. Satake I., Linear algebra, Shokabo, Tokyo, 1989 (in Japanese).
  5. Fujii K., Beyond the Gaussian. II. Some applications, in progress.
  6. Morozov A., Shakirov Sh., New and old results in resultant theory, Theoret. and Math. Phys. 163 (2010), 587-617, arXiv:0911.5278.
  7. Dolotin V., Morozov A., Introduction to non-linear algebra, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, hep-th/0609022.

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