Abstract
Let \({\textnormal {R}}\) be a real closed field, \(\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]\) finite subsets of polynomials, with the degrees of the polynomials in \(\mathcal{P}\) (resp., \(\mathcal{Q}\)) bounded by d (resp., d 0). Let \(V \subset {\textnormal {R}}^{k}\) be the real algebraic variety defined by the polynomials in \(\mathcal{Q}\) and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family \(\mathcal{P}\) on V is bounded by
where \(s = \operatorname {card}\mathcal{P}\), and
In case 2d 0≤d, the above bound can be written simply as
(in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d 0≪d) the best known bound of
on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d 0.
The distinction between the bound d 0 on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in \(\mathcal{P}\) that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).
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Barone, S., Basu, S. Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety. Discrete Comput Geom 47, 577–597 (2012). https://doi.org/10.1007/s00454-011-9391-3
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DOI: https://doi.org/10.1007/s00454-011-9391-3