Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets

In [8] counting complexity classes #P R and #P C in the Blum-Shub-Smale (BSS) setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [8] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the cla...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 5; no. 4; pp. 351 - 387
Main Authors Bürgisser, Peter, Cucker, Felipe, Lotz, Martin
Format Journal Article
LanguageEnglish
Published New York Springer 01.11.2005
Springer Nature B.V
Subjects
Algebra
Complex numbers
Complexity
Complexity (Philosophy)
Computational physics
Mathematical physics
Mathematics
Numbers
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Summary:In [8] counting complexity classes #P R and #P C in the Blum-Shub-Smale (BSS) setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [8] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FP R #PR . In this paper, we prove that the corresponding result is true over C, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FP C #PC . We also obtain a corresponding completeness result for the Turing model. [PUBLICATION ABSTRACT]
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-005-0146-x