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Whitney number of closed real algebraic affine curve of type I. (English) Zbl 1131.14062
A real plane irreducible algebraic curve $$A$$ is called of type I if its complement in the complexification $${\mathbb C}A\backslash A$$ is not connected and, thus, consists of two components. Choosing one of them, $${\mathbb C}A_+$$, one induces a so-called complex orientation on $$A$$. Given a real line $$L$$, one similarly picks up a component $${\mathbb C}L_+$$ of $${\mathbb C}L\backslash L$$ and respectively orients $$L$$. Under assumptions that $$A\backslash L$$ is compact and has only simple real nodes (transverse intersections of smooth real branches) as singularities, one can define the Whitney number $$w(A)$$ as the degree of the Gauss map $$A\to L$$. The main result states that $$w(A)={\mathbb C}A_+\circ{\mathbb C}L_+-{\mathbb C}A_+\circ{\mathbb C}L_-$$. A simple elegant proof is based on the idea to rotate $$L$$ around one of its points and express $$w(A)$$ as the signed number of tangencies of the rotating line with $$A$$.

##### MSC:
 14P25 Topology of real algebraic varieties 57R42 Immersions in differential topology 53A99 Classical differential geometry
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