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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 [Ber] theorem. One may, in addition, require that all roots lie in a prescribed open subset of the solution space. This note characterizes the maximal number of regular real roots of systems of polynomial equations that are multihomogeneous, in the sense that the variables of the system are partitioned into groups, with each equation being homogeneous separately in the variables of each group. Examples of multihomogeneous equations include the (singly) homogeneous equations considered in Bezout's theorem, multilinear functions, and the equations characterizing totally mixed Nash equilibria of a normal form game. The general form of a sparse system of d polynomial equations in d variables is f(x) = (f 1 (x); : : : ; f d (x)) = 0; (1) where x = (x 1; : : : ; x d) and, for each i = 1; : : : ; d, there is a nonempty finite A i ae N d, |
