# The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations

1999-03

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The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations

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1999-03

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Center for Economic Research, Department of Economics, University of Minnesota

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Working Paper

## Abstract

The methods of Shub and Smale [SS93] are extended to the class of multihomogeneous systems
of polynomial equations, yielding Theorem 1, which is a formula expressing the mean (with respect to a
particular distribution on the space of coefficient vectors) number of real roots as a multiple of the mean
absolute value of the determinant of a random matrix. Theorem 2 derives closed form expressions for the
mean in special cases that include: (a) Shub and Smale's result that the expected number of real roots
of the general homogeneous system is the square root of the generic number of complex roots given by Bezout's theorem; (b) Rojas' [Roj96] characterization of the mean number of real roots of an "unmixed"
multihomogeneous system. Theorem 3 gives upper and lower bounds for the mean number of roots, where
the lower bound is the square root of the generic number of complex roots, as determined by Bernstein's
[Ber75] theorem. These bounds are derived by induction from recursive inequalities given in Theorem 4.

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## Previously Published Citation

McLennan, A., (1999), "The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations", Discussion Paper No. 307, Center for Economic Research, Department of Economics, University of Minnesota.

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McLennan, Andrew. (1999). The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/55857.

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