**Beiträge zur Algebra und Geometrie <BR> Contributions to Algebra and GeometryVol. 40, No. 2, pp. 343-350 (1999)
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The Maximum Number of Real Roots of a Multihomogeneous System of Polynomial Equations

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Andrew McLennan

Department of Economics, University of Minnesota, 271 19th Avenue South, Minneapolis, MN 55455 e-mail: mclennan@atlas.socsci.umn.edu

**Abstract:** A multihomogeneous system of polynomial equations, with as many equations as degrees of freedom, has instances for which there are as many regular real roots, in the relevant product of projective spaces, as are allowed, for the corresponding dehomogenized system, by Bernshtein's [B] theorem. One may, in addition, require that all roots lie in a prescribed open subset of the solution space. The maximal number of roots is characterized as a multiple (by the inverse of a product of factorials) of the permanent of a certain matrix, and recursive formulas are given for this number.

\item{[B]} Bernshtein, D. N.: * The number of roots of a system of equations*. Functional Analysis and its Applications ** 9** (1975), 388-391.

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