Minimizing multi-homogeneous Bézout numbers by a local search method
This paper is about the preparations for computing the zeros of polynomial systems via a homotopy method. For this it is important that the number of homotopy paths is as small as possible. This number is bounded by the Bézout and the multi-homogeneous Bézout numbers. For the latter it is important to partition the variables into disjoint classes. The authors develop a ``topology'' on these classes, and starting from a certain point proceed to improve the multi-homogeneous Bézout number through local searches. They demonstrate the effectiveness with a few examples and conjecture that the problem of finding the minimal multi-homogeneous Bézout number may be NP-hard.
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