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Mappings of real algebraic hypersurfaces. (English) Zbl 0869.14025
A smooth real hypersurface in $$\mathbb{C}^N$$ is called algebraic if it is contained in the zero set of a nonzero real-valued polynomial in $$Z$$ and $$\overline Z$$. A hypersurface $$M$$ in $$\mathbb{C}^N$$ is holomorphically degenerate at a point $$p_0 \in M$$ if there exists a nonzero germ of a holomorphic vector field tangent to $$M$$ in a neighborhood of $$p_0$$. The paper contains a complete characterization of algebraic hypersurfaces in $$\mathbb{C}^N$$ such that any holomorphic map with non-vanishing Jacobian determinant between two such hypersurfaces is algebraic. Let $$M$$ be a connected algebraic hypersurface in $$\mathbb{C}^N$$, $$N>1$$, and let $$p_0$$ be a point of $$M$$. Every biholomorphism defined in a neighborhood of $$p_0$$ and mapping $$M$$ to another algebraic hypersurface in $$\mathbb{C}^N$$ is algebraic if and only if $$M$$ is not holomorphically degenerate at any point. This is a generalization of a result of S. M. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)].
The authors also describe a relationship between essential finiteness and holomorphic nondegeneracy. If $$M$$ is a connected real analytic hypersurface in $$\mathbb{C}^N$$, then there exists a point of $$M$$ in which $$M$$ is essentially finite if and only if $$M$$ is not holomorphically degenerate at any point in $$M$$.

##### MSC:
 14P99 Real algebraic and real-analytic geometry 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 14J70 Hypersurfaces and algebraic geometry
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