计算数值计算的复杂性类II：代数和半代数集
Counting complexity classes for numeric computations II: algebraic and semialgebraic sets
摘要：
我们定义计数类#P_R和#P_C在Blum-Shub-Smale设置的计算实数或复数，分别。分别计算R上多项式不等式系统或C上多项式等式系统的解的个数的问题，在这些类中是自然完备的问题。我们调查在何种程度上新的计数类捕捉计算基本拓扑不变量的半代数集（在R）和代数集（在C）的复杂性。证明了半代数集的（修正）欧拉特征线的计算问题是FP_R^{#P_R}-完全的，复代数集的几何次数的计算问题是FP_C^{#P_C}-完全的.在经典图灵模型中，通过取其布尔部分，定义了新的计数复杂性类，并证明了由整数多项式给出的（半）代数集的欧拉特征和几何次数的计算问题在这些类中是完备的.我们补充的图灵模型的结果证明，对于所有的k在N中，FPSPACE-硬度的问题计算的第k个贝蒂数zet的实零点的一个给定的整数多项式。这适用于奇异同调和Borel-Moore同调。
英文摘要：
We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the (modified) Euler characteristic of semialgebraic sets is FP_R^{#P_R}-complete, and that the problem of computing the geometric degree of complex algebraic sets is FP_C^{#P_C}-complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k in N, the FPSPACE-hardness of the problem of computing the k-th Betti number of the zet of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.
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