Bounding the equivariant Betti numbers of symmetric semi-algebraic sets

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• 作者：Saugata Basu^a ; ¹ ; ^{sbasu@math.purdue.edu" class="auth_mail" title="E-mail the corresponding author} ; Cordian Riener^b ; ^{cordian.riener@aalto.fi" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Equivariant Betti numbers ; Semi-algebraic sets ; Symmetric polynomials
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：803-855
• 全文大小：898 K

文摘

Let class="boldFont">R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si1.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=727e4c18dcd7f5deaa84d2e89e4b3d72" title="Click to view the MathML source">R^kclass="mathContainer hidden">class="mathCode">${\mathbf{R}}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Oleĭnik and Petrovskiĭ, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant   Betti numbers of symmetric algebraic and semi-algebraic subsets of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si1.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=727e4c18dcd7f5deaa84d2e89e4b3d72" title="Click to view the MathML source">R^kclass="mathContainer hidden">class="mathCode">${\mathbf{R}}^k$. We obtain several asymptotically tight upper bounds. In particular, we prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si3.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=77c1da6865a09c8a6ad1754493c31be3" title="Click to view the MathML source">S⊂R^kclass="mathContainer hidden">class="mathCode">$S\subset {\mathbf{R}}^k$ is a semi-algebraic subset defined by a finite set of s symmetric polynomials of degree at most d  , then the sum of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si4.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=12fc409c1fdb361800c7da6fe1e19534" title="Click to view the MathML source">S_kclass="mathContainer hidden">class="mathCode">${\mathfrak{S}}_k$-equivariant Betti numbers of S   with coefficients in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si5.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=edbbe0d4e5f572afe5080ddf42a209ee" title="Click to view the MathML source">Qclass="mathContainer hidden">class="mathCode">$\mathbb{Q}$ is bounded by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816312191&_mathId=si6.gif&_user=111111111&_pii=S0001870816312191&_rdoc=1&_issn=00018708&md5=aa5592a6b2f03a39fbfe0633e6450433" title="Click to view the MathML source">(skd)^O(d)class="mathContainer hidden">class="mathCode">${(skd)}^{O(d)}$. Unlike the classical bounds on the ordinary Betti numbers of 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. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.