Bounding the equivariant Betti numbers and computing the generalized Euler-Poincaré characteristic of symmetric semi-algebraic sets

Unlike the well known classical bounds due to Oleinik and Petrovskii, Thom and Milnor on the Betti numbers of (possibly non-symmetric) real algebraic varieties and semialgebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. Moreover, our bounds are asymptotically tight. As an application we improve the best known bound on the Betti numbers of the projection of a compact semi-algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.