[formula omitted]-norm bounds and metric properties for zero loci of real analytic functions.

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. [ABSTRACT FROM AUTHOR]