Localization of real algebraic hypersurfaces with applications to the enumeration of the classes of relative equilibria of a (5+1)-body problem

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:485

Article

Barros, Jean F.¹  Leandro, Eduardo S. G.² 

[1] Univ Estadual Feira de Santana, Dept Ciencias Exatas, Av Transnor Destina S-N, BR-44036900 Feira De Santana, BA, Brazil

[2] Univ Fed Pernambuco, Dept Matemat, Av Jornalista Anibal Fernandes S-N, BR-50740560 Recife, PE, Brazil

关键词: Algebraic hypersurfaces;    N-body problem;    Relative equilibria;   

DOI  :  10.1016/j.jmaa.2019.123813

来源: Elsevier

【 摘 要 】

We extend classical results on the localization of zeros of real univariate polynomials to the localization of zero sets of real multivariate polynomials P, more precisely, of real algebraic hypersurfaces (assuming 0 is a regular value). Through suitable changes of variables, we may verify whether such a hypersurface P-1(0) in R-n intersects or not a given n-dimensional box B-n = Pi(n)(l=1)[a(l), b(l)], and in the affirmative case, to locate with arbitrary precision the set P-1(0) boolean AND B-n Properties of the hypersurface such as being an analytic graph may also be deduced from our results, which include a non-differentiable, non-local version of the implicit function theorem for polynomials. Next, we apply the ideas of the first part to study the bifurcations of a one-parameter family of symmetric classes of relative equilibria of the (5 + 1)-body problem. The exact numbers of classes of relative equilibria are provided, and our technique allows for the localization of all relative equilibria. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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