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Computer Science > Computational Complexity

arXiv:cs/0405021 (cs)
[Submitted on 5 May 2004]

Title:Computing Multi-Homogeneous Bezout Numbers is Hard

Authors:Gregorio Malajovich, Klaus Meer
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Abstract: 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 contains NP.
Subjects: Computational Complexity (cs.CC); Symbolic Computation (cs.SC)
ACM classes: F.2.1;G.1.5
Cite as: arXiv:cs/0405021 [cs.CC]
  (or arXiv:cs/0405021v1 [cs.CC] for this version)
  https://doi.org/10.48550/arXiv.cs/0405021
Journal reference: Theory of Computing Systems, Volume 40, Number 4 / June, 2007
Related DOI: https://doi.org/10.1007/s00224-006-1322-y

Submission history

From: Gregorio Malajovich [view email]
[v1] Wed, 5 May 2004 14:33:19 UTC (24 KB)
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