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Session Real algebraic geometry, applications and related topics
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Bounds on equivariant Betti numbers for symmetric semi algebraic sets |
Cordian Riener, Aalto University, Finland
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The talk is based on the joint work with Saugata Basu
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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.
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