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

McLennan, Andrew (Center for Economic Research, Department of Economics, University of Minnesota, 1999)

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

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

Working Paper

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.

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.

Discussion Paper

307

McLennan, Andrew.
(1999).
The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations.
Center for Economic Research, Department of Economics, University of Minnesota.
Retrieved from the University of Minnesota Digital Conservancy,
http://hdl.handle.net/11299/55857.

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