A criterion for a proper rational map to be equivalent to a proper polynomial map

Let B be the unit ball in the complex space C. Write Rat(B,B) for the space of proper rational holomorphic maps from B into B and Poly(B,B ) for the set of proper polynomial holomorphic maps from B into B . We say that F and G ∈ Rat(B,B) are equivalent if there are automorphisms σ ∈ Aut(B) and τ ∈ Aut(B ) such that F = τ ◦G◦σ. Proper rational holomorphic maps from B into B with N ≤ 2n− 2 are equivalent to the identity map ([Fa] [Hu]). In [HJX], it is shown that F ∈ Rat(B,B ) with N ≤ 3n−4 is equivalent to a quadratic monomial map, called the D’Angelo map. However, when the codimension is sufficiently large, there are a plenty of rooms to construct rational holomorphic maps with certain arbitrariness by the work in Catlin-D’Angelo [CD]. Hence, it is reasonable to believe that after lifting the codimension restriction, many proper rational holomorphic maps are not equivalent to polynomial proper holomorphic maps. In the last paragraph of the paper [DA], D’Angelo gave a philosophic discussion on this matter. However, explicit examples of proper rational holomorphic maps, that are not equivalent to polynomial proper holomorphic maps, do not seem to exist in the literature. And the problem of determining if an explicit proper rational holomorphic map is equivalent to a polynomial holomorphic map does not seem to have been studied so far. This short paper is concerned with such a problem. We will first give an explicit criterion when a rational holomorphic map is equivalent to a polynomial holomorphic map. Making