由几个二次不等式定义的半代数集的投影
On Projections of Semi-algebraic Sets Defined by Few Quadratic Inequalities
摘要：
令$S \subset \R^{k + m}$是由次数最多为2的$S \subset \R^{k + m}$多项式不等式系统定义的紧致半代数集。$ 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 \ell）}. $ We also present an algorithm for computing the the first $ q $ Betti numbers of $ π（S）$, whose complexity is $（k+m）^{2^{O（q \ell）}}。$ For fixed $ q $ and $ \ell $, both the bounds are polynomial in $ k+m$。
英文摘要：
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 $π$ 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 $(k + m)^{O(q\ell)}.$ We also present an algorithm for computing the the first $q$ Betti numbers of $π(S)$, 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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