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On the classification of rational sphere maps. (English) Zbl 1381.32008
The author considers the problem of classifying (CR) rational maps between unit spheres in complex spaces. For this purpose, Hermitian forms are the main employed tool. The author defines so-called ancestors and descendants of Hermitian forms associated with the rational sphere map under study. In particular, he shows that every rational sphere map is an ancestor of a final descendant. When the denominator of such a rational sphere map is of first degree, the author also provides a decisive result concerning the classification of possible numerators of the mentioned final descendant. Moreover, the author finds that, if $$f$$ is the final descendant of a certain rational map with the associated Hermitian forms invariant under the circle action $$z\mapsto \exp(i\theta)\, z$$, then $$f$$ is either a polynomial or its numerator and denominator have the same degree.
It should be worth to state that in Section 2, the author also provides a nice and satisfactory description of the results achieved up to the date of preparing this paper, which may help a lot audiences which are interested in the subject.

##### MSC:
 32H35 Proper holomorphic mappings, finiteness theorems
##### Keywords:
rational sphere maps; Hermitian forms
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