Let $R$ be a real closed field. We prove upper bounds on the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of $R^k$. More precisely, we prove that if $S\subset R^k$ is a semi-algebraic subset defined by a finite set of $s$ symmetric polynomials of degree at most $d$, then the sum of the $S_k$ equivariant Betti numbers of $S$ with coefficients in $\mathbf{Q}$ is bounded by $s^5d(kd)^{O(d)}$. 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 semi-algebraic 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. (Joint work with Saugata Basu)
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