Cordian Riener: Betti numbers of symmetric semialgebraic sets

Semi-algebraic sets are subsets of R^k that can be described using inequalities. There has been a lot of work on bounding the Betti numbers of such sets. Starting with classical work by Tom, Milnor, Ole{\u\i}nik and Petrovski{\u\i}. Motivated by algorithmic results, I will present results in the setting of symmetric (i.e. S_k-invariant) semi algebraic sets defined by polynomials of degree d. In this setting I will show a bound on the equivariant Betti numbers. Unlike the well known classical bounds due algebraic varieties and semi-algebraic sets this bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant.

Moreover, our bounds are asymptotically tight. I will further sketch how this generalizes to the isotypic decomposition of the homology groups.  

As an application the bounds given can be used to improve the best known bound on the Betti numbers of the projection of a compact semi-algebraic (not necessary symmetric) set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.