Topological invariants of analytic sets associated with Noetherian families. (Q1777625)

This paper is devoted to represent the Euler characteristic of the intersections of certain analytic germs with spheres of small enough radius in terms of the sums of signs of certain finite families of analytic functions germs. More precisely, let \(\Omega \subset {\mathbb R}^n\) be a compact semianalytic set and \({\mathcal F}\) a collection of real analytic functions defined in some neighbourhood of \(\Omega\). For each \(\omega\in \Omega\), the analytic germs \(Y_{\omega}=\bigcap_{f\in{\mathcal F}}f^{-1}(0)\) at \(\omega\) and \(X_{\omega}=Y_\omega-\omega\) are considered. Using similar arguments to the ones in [\textit{A. Parusiński, Z. Szafraniec}, Manuscr. Math. 93, No. 4, 443--456 (1997; Zbl 0913.14019)] the author proves that there exist analytic functions \(v_1,\ldots,v_s\) defined in a neighbourhood of \(\Omega\) such that for each \(\omega\in\Omega\) there exists \(0<\varepsilon_\omega\gg 1\) such that for each \(0<\varepsilon<\varepsilon_\omega\) \[ \frac{1}{2}\chi(S_{\varepsilon}^{n-1}\cap X_\omega)=\sum_{i=1}^s\text{sgn} v_i(\omega), \] where \(S_{\varepsilon}^{n-1}\) is the sphere of \({\mathbb R}^n\) of center the origin and radius \(\varepsilon\). The previous result also holds taking as \({\mathcal F}\) a family of analytic functions from an \(\Omega\)-Noetherian algebra satisfying some additional conditions (see Section 4). Finally, in section 5, the author presents certain relevant consequences of the main result of the article which is the one stated above.