Polynomial and rational maps between balls (Q612269)

Let \(\mathbb B_n\) be the unit ball in \(\mathbb C^n\). We say that the rational holomorphic maps \(f,g:\mathbb B_n\rightarrow\mathbb B_m\), \(n,m\in\mathbb N\), are equivalent if there are automorphisms \(\sigma:\mathbb B_n\rightarrow\mathbb B_n\), \(\tau:\mathbb B_m\rightarrow\mathbb B_m\) such that \(f=\tau\circ g\circ\sigma\). In the paper the authors give a criterion when a nonconstant rational holomorphic map \(f:\mathbb B_n\rightarrow\mathbb B_m\), \(n,m\in\mathbb N\), is equivalent to a polynomial holomorphic map \(g:\mathbb B_n\rightarrow\mathbb B_m\). Next, they use that criterion to show that any rational map \(f:\mathbb B_2\rightarrow\mathbb B_m\) of degree two is equivalent to a polynomial proper holomorphic map \(g:\mathbb B_2\rightarrow\mathbb B_m\). Moreover, they give an example of rational holomorphic maps of degree three that are ''almost'' linear but are not equivalent to holomorphic polynomial maps.