On {\L}ojasiewicz Ideals and Flatness for Zero Sets with Infinite Tangential Geometry

Abstract

Let $\Omega \subset \mathbb{R}^n $ be an open set, and let $\mathcal{E}(\Omega)$ be the ring of infinitely differentiable functions on $\Omega$. For an ideal $I \subset \mathcal{E}(\Omega)$, we denote by $Z(I)$ its zero set. A classical result of Ren\'e Thom asserts that if $I$ is a finitely generated {\L}ojasiewicz ideal, then $Z(I)$ contains an open dense subset of smooth points. The goal of this note is to examine a converse question: does the existence of an open dense set of smooth points in $Z(I)$ ( $I\subset\mathcal{E}(\Omega) $ is a finitely generated ideal) imply that the ideal $I$ is \L{}ojasiewicz? We analyze obstructions to such a converse and identify geometric conditions under which it fails. As an application, we study smooth real-valued functions of two variables whose zero set coincides with the classical Hawaiian earring, the union of infinitely many tangent circles accumulating at the origin. We show that any such function must be flat at the origin, in the sense that all partial derivatives vanish there. We formulate a geometric criterion concerning families of smooth arcs tangent at a point with unbounded or infinitely varying curvature, and derive from it a degenerate form of the {\L}ojasiewicz inequality adapted to this non-analytic setting.