LAUSR

Abstract 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 = Π l = 1 n [ a l , b l ] , and in the affirmative case, to locate with arbitrary precision the set P − 1 ( 0 ) ∩ 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.