Polynomial and Rational Maps between Balls
Polynomial and Rational Maps between Balls
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DOI：
10.4310/pamq.2010.v6.n3.a10
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发表时间：
2010
影响因子：
0.7
通讯作者：
J. Faran;Xiaojun Huang;S. Ji;Y. Zhang
中科院分区：
文献类型：
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作者：
J. Faran;Xiaojun Huang;S. Ji;Y. Zhang
设B是复空间C中的单位球。记Rat（B，B）为从B到B的真有理全纯映射的空间，Poly（B，B）为从B到B的真全纯多项式映射的集合。如果存在自同构σ ∈ Aut（B）和τ ∈ Aut（B）使得F = τ o G o σ，则我们说F和G ∈ Rat（B，B）是等价的。从B到B且N ≤ 2n − 2的真全纯映射，如果在边界上足够光滑，则等价于恒等映射（[Fa] [Fr] [Hu]）。在[HJX]中，证明了当N ≤ 3 n − 4时，F ∈ Rat（B，B）等价于一个二次单项映射，称为D 'Angelo映射。然而，当余维数足够大时，Catlin-D 'Angelo [CD]的工作为构造具有一定任意性的有理全纯映射提供了很大的空间。因此，我们有理由相信，在提升余维限制后，许多真有理全纯映射不等价于真全纯多项式映射。在最后一段的文件[DA]，德安杰洛了一个哲学讨论这个问题。然而，确定一个显式真有理全纯映射是否等价于一个多项式映射的问题似乎还没有被研究过。这篇简短的论文就涉及到这样一个问题。我们首先给出有理全纯映射等价于全纯多项式映射的一个简单而明确的判据。借助于[CJX]中的分类结果，在§3中利用这个准则证明了从B到B的二次真有理全纯映射等价于
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