Efficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets

Abstract

Let $\mathrm{R}$ be a real closed field and $\mathrm{D}\subset \mathrm{R}$ 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 ${\mathrm{R}}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincaré characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, 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 ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.

29 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1312.6582

Cited by in corpus (1)

• Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets