Theory of Computing Systems

, Volume 40, Issue 4, pp 553–570 | Cite as

Computing Minimal Multi-Homogeneous Bezout Numbers Is Hard

  • Gregorio MalajovichEmail author
  • Klaus MeerEmail author


The multi-homogeneous Bezout number is 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 Bezout number. In this paper it is proved that the problem of computing, or even estimating, the optimal multi-homogeneous Bezout number is actually 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. Moreover, polynomial-time algorithms for estimating the minimal multi-homogeneous Bezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP ⫆ NP.


Polynomial Time Convex Body Combinatorial Optimization Problem Polynomial System Suitable Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Departamento de Matematica Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970Rio de Janeiro, RJBrasil
  2. 2.Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230Odense M.Denmark

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