Computing the Homology of Real Projective Sets
Computing the Homology of Real Projective Sets
复制标题
计算实射影集的同调
DOI：
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发表时间：
2016
影响因子：
3
通讯作者：
M. Shub
中科院分区：
文献类型：
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作者：
F. Cucker;T. Krick;M. Shub
我们描述并分析了一种用于计算实射影簇同源性（贝蒂数和挠系数）的数值算法。这里数值意味着算法在数值上是稳定的（在某种意义上是精确的）。其成本取决于输入的条件及其大小，并且在变量数量（环境空间的维度）和条件中的多项式以及定义多项式的次数方面呈单指数。此外，我们还表明，除了数据量呈指数级小的一组特殊度量之外，该算法需要指数级时间。
We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.