Abstract
Computing good estimates for the number of solutions of a polynomial system is of great importance in many areas such as computational geometry, algebraic geometry, mechanical engineering, to mention a few. One prominent and frequently used example of such a bound is the multi-homogeneous Bézout number. It provides a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bézout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous Bézout number is NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. As a consequence, polynomial time algorithms for estimating the minimal multi-homogeneous Bézout number up to a fixed factor cannot exist even in a randomized setting, unless BPP⊒NP.
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Malajovich, G., Meer, K. (2005). Computing Minimal Multi-homogeneous Bézout Numbers Is Hard. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_20
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DOI: https://doi.org/10.1007/978-3-540-31856-9_20
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