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

We give algorithms with polynomially bounded complexities (for fixed degrees) for computing the generalized Euler-Poincaré characteristic of semi-algebraic sets defined by symmetric polynomials. This is in contrast to the best complexity of the known algorithms for the same problem in the non-symmetric situation, which is 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.