On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities |
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Authors: | Saugata Basu Thierry Zell |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA |
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Abstract: | Let S⊂ℝ
k+m
be a compact semi-algebraic set defined by P
1≥0,…,P
ℓ
≥0, where P
i
∈ℝX
1,…,X
k
,Y
1,…,Y
m
], and deg (P
i
)≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ
k+m
onto ℝ
m
. We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m)
O(q
ℓ). We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is
. For fixed q and ℓ, both the bounds are polynomial in k+m.
The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship. |
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Keywords: | Betti numbers Quadratic inequalities Semi-algebraic sets Spectral sequences Cohomological descent |
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