We consider the problem of deciding whether or not a zero locus, X, of multivariate
real analytic functions crosses a given r-norm ball in the real n-dimensional affine
space. We perform a local study of the problem, and we provide both necessary and
sufficient conditions to answer the question. Our conditions derive from the analysis
of differential geometric properties of X at the center of the ball. An algorithm to
evaluate r-norms distances is proposed.Ministerio de Economía y CompetitividadEuropean Regional Development FundA major part of this work was developed while J.R. Sendra was visiting, in the frame of GNSAGA—Istituto Nazionale di Alta Matematica, the University of Genova, and while M.C. Beltrametti and M. Torrente were visiting the University
of Alcalá, in the frame of the project Giner de los Rios and of GNSAGA—Istituto Nazionale di Alta Matematica
