## 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 R^{k}, 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 language | English |
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Title of host publication | Contemporary Mathematics |

Publisher | AMERICAN MATHEMATICAL SOCIETY |

Pages | 51-79 |

Number of pages | 29 |

Volume | 697 |

ISBN (Electronic) | 978-1-4704-4222-4 |

ISBN (Print) | 978-1-4704-2966-9 |

Publication status | Published - 2017 |

MoE publication type | A3 Part of a book or another research book |