Efficient algorithms for computing the euler-poincaré characteristic of symmetric semi-algebraic sets

Saugata Basu, Cordian Riener

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

2 Citations (Scopus)

Abstract

Let R be a real closed field and D ⊂ R an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincaré characteristic of real algebraic as well as semi-algebraic subsets of Rk, which are defined by symmetric polynomials with coefficients in D. We give algorithms for computing the generalized Euler-Poincaré characteristic of such sets, whose complexities measured by the number of arithmetic operations in D, are polynomially bounded in terms of k and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result (#P-hardness) coming from discrete complexity theory.

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAMERICAN MATHEMATICAL SOCIETY
Pages51-79
Number of pages29
Volume697
ISBN (Electronic)978-1-4704-4222-4
ISBN (Print)978-1-4704-2966-9
Publication statusPublished - 2017
MoE publication typeA3 Part of a book or another research book

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