Real Algebraic and Analytic Geometry

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84. Aleksandra Nowel:
Topological invariants of analytic sets associated with Noetherian families.


Submission: 2004, January 29.

Let $\Omega$ be a Noetherian space, $A(\Omega)$ -- the corresponding algebra, and let $X_{\omega}\subset (R^n,0)$ ($\omega\in\Omega$) be a Noetherian family of analytic germs. Let $g(\omega)=\frac{1}{2}\chi(X_{\omega}\cap S_{\epsilon})$, where $\chi(X_{\omega}\cap S_{\epsilon})$ denotes the Euler characteristic of the intersection of $X_{\omega}$ with a sphere of small radius $\epsilon$. Then there exists a finite family $v_1 ,\ldots ,v_s \in A(\Omega)$ such that $$g(\omega)=\sum_{i=1}^{s} \sgn v_i(\omega).$$.

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