부산대학교 도서관

Let $U$ be a convex open subset of $\Bbb R^n$ and let $f_1,\ldots,f_m \: U \to \Bbb R$, for $m \le n$, be real analytic functions. Let $B_r(p,\epsilon)$ denote the ball with center $p$ and radius $\epsilon$ with respect to the $r$-norm $\|x\|_r = (\sum_{i = 1}^n |x_i|^r)^{1/r}$, where $r \in [1,\infty)$ and $\|x\|_\infty = \max_i |x_i|$. In this paper the authors give necessary and sufficient conditions for the non-trivial intersection of the common zero locus of the $f_i$ and the ball $B_r(p,\epsilon)$ (provided that the ball is contained in $U$). The proofs are based on Taylor's formula and the Moore-Penrose pseudo-inverse (of the Jacobian of $f=(f_1,\ldots,f_m)$). To analyze the inconclusive regime, a heuristic algorithm is proposed. Several numerical examples illustrate the results of the paper. The article is a continuation of the work of M.-L. Torrente and M.~C. Beltrametti [J. Algebra Appl. {\bf 13} (2014), no.~8, 1450057; MR3225124] and E. Saggini and Torrente [J. Algebr. Stat. {\bf 7} (2016), no.~1, 45--71; MR3529334] in which the algebraic hypersurface case and the cases $r =2$ and $r = \infty$ were considered.