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Computing Multi-Homogeneous Bezout Numbers is Hard

The multi-homogeneous Bézout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations. Estimating the number of isolated solutions of a polynomial system is useful for the design and analysis of homotopy algorithms and certain applications [2, 5]. An application of multi-homogeneous Bézout bounds outside the realm of algebraic equation solving is discussed in [3], where the number of roots is used to bound geometrical quantities such as volume and curvature. There is an important connection between rootcounting and NP-completeness theory. Indeed, it is easy to reduce an NP-complete or NP-hard problem such as SAT, the Traveling Salesman problem, Integer Programming (and thus all other NP problems as well) to the question whether certain polynomial systems have a common zero.