摘要翻译：
在这篇论文中，我们考虑由二次多项式定义的实闭域$R$上的半代数集。R^k$的半代数集定义为R^k$中既包含代数集又包含多项式不等式定义的集，且在布尔运算（补、有限并、有限交）下闭的最小集族。在由二次多项式定义的实闭域$R$上，利用定义它们的多项式系统的参数，证明了半代数集的Betti数和某些纤维的不同稳定同伦类型的个数的新界，改进了已有的结果。最后，我们给出了两个新的算法及其实现。第一种算法计算由$\mathbb{R}^k$中的紧致对象定义的半代数集的连通分量数和第一Betti数。该算法改进了已知的半代数集三角剖分方法。此外，该算法还得到了以前不可能实现的高效实现。第二种算法用半数值方法有效地计算$\mathbb{R}^3$中三个二次曲面的实交。
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英文标题：
《Algorithmic and topological aspects of semi-algebraic sets defined by
  quadratic polynomial》
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作者：
Michael Kettner
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Computer Science        计算机科学
二级分类：Computational Geometry        计算几何
分类描述：Roughly includes material in ACM Subject Classes I.3.5 and F.2.2.
大致包括ACM课程I.3.5和F.2.2中的材料。
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一级分类：Mathematics        数学
二级分类：Algebraic Topology        代数拓扑
分类描述：Homotopy theory, homological algebra, algebraic treatments of manifolds
同伦理论，同调代数，流形的代数处理
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一级分类：Mathematics        数学
二级分类：Geometric Topology        几何拓扑
分类描述：Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
流形，轨道，多面体，细胞复合体，叶状，几何结构
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英文摘要：
  In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.
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PDF链接：
https://arxiv.org/pdf/0709.3283