On the Roots of a Random System of Equations. The Theorem of Shub and Smale and Some Extensions |
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| Authors: | Jean-Marc Azaïs Mario Wschebor |
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| Institution: | (1) Laboratoire de Statistique et Probabilites, UMR-CNRS C5583 Universite Paul Sabatier, 118, route de Narbonne 31062 Toulouse Cedex 4, France;(2) Centro de Matematica, Facultad de Ciencias, Universidad de la Republica, Calle Igua 4225, 11400 Montevideo, Uruguay |
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| Abstract: | We give a new proof of a theorem of Shub and Smale on the expectation of the number of roots of a system of m random
polynomial equations in m real variables, having a special isotropic Gaussian distribution. Further, we present a certain number
of extensions, including the behavior as m → +∞ of the variance of the number of roots, when the system of equations is also stationary. |
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| Keywords: | Random polynomials System of random equations Rice
formula |
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