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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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