Consider the rational mappings from into of the form , where and is an onto complex linear map. Two such mappings are equivalent, if they are equal after biholomorphic transformations of the domain and target, respectively.
The author gives a complete classification for the equivalence problem. For the proof, the author associates to each mapping a complex matrix , and shows that the equivalence of two such mappings implies the similarity of the corresponding matrices under matrix transformations satisfying a reality condition. The converse is also true, which is however a consequence of the geometric interpretation of the normal form of the matrix . The complete set of equivalence classes of mappings consists of a continuous family of embeddings of in with exactly two elliptic complex tangents, a continuous family of totally real immersions of in with one point of self-intersection, and finitely many others.
The author also studies the equivalence problem for analogous mappings from into . The complete classification for this case and for the higher dimension case is unsettled in this paper. However, the author gives some interesting examples.