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

TL;DR

This paper establishes an explicit criterion to determine when a rational holomorphic map between balls is equivalent to a polynomial map, and applies it to classify certain low-degree maps.

Contribution

It provides a new explicit criterion for equivalence and classifies degree 2 maps, also constructing degree 3 examples that are not equivalent to polynomials.

Findings

Any degree 2 proper rational map from B^2 to B^N is equivalent to a polynomial map.

Constructed degree 3 rational maps that are almost linear but not equivalent to polynomials.

Abstract

In this paper, we give an explicit criterion when a rational holomorphic map between balls is equivalent to a polynomial holomorphic map. Making use of this criterion, we show that any proper rational holomorphic map from B^2 into B^N of degree two is equivalent to a polynomial holomorphic map; we also construct rational holomorphic maps of degree 3 that are almost linear but are not equivalent to polynomial holomorphic maps.