On Projections of Semi-algebraic Sets Defined by Few Quadratic Inequalities

Details On Projections of Semi-algebraic Sets Defined by Few Quadratic Inequalities On Projections of Semi-algebraic Sets Defined by Few Quadratic Inequalities

2006-02-17

arxiv.org/abs/math/0602398v1

Discrete and Computational Geometry 39 (2008), 100--122.

Mathematics

Algebraic Geometry

21 pages, 1 figure. Requires diagrams.tex

Scientific paper

10.1007/s00454-007-9014-1

Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $ Let $\pi$ 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 $\pi(S)$ is bounded by $(k + m)^{O(q\ell)}.$ We also present an algorithm for computing the the first $q$ Betti numbers of $\pi(S)$, whose complexity is $ (k+m)^{2^{O(q\ell)}}.$ For fixed $q$ and $\ell$, both the bounds are polynomial in $k+m$.

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