Polynomial and Rational Maps between Balls

Abstract

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 holomorphic polynomial 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 holomorphic maps from B into B with N ≤ 2n − 2, that are sufficiently smooth up to the boundary, are equivalent to the identity map ([Fa] [Fr] [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 is plenty of room 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 proper holomorphic polynomial maps. In the last paragraph of the paper [DA], D’Angelo gave a philosophic discussion on this matter. However, the problem of determining if an explicit proper rational holomorphic map is equivalent to a polynomial map does not seem to have been studied so far. This short paper is concerned with such a problem. We will first give a simple and explicit criterion when a rational holomorphic map is equivalent to a holomorphic polynomial map. With the help of the classification result in [CJX], this criterion is used in §3 to show that proper rational holomorphic maps from B into B of degree two are equivalent to