Exhaustion functions and normal forms for proper maps of balls

TL;DR

This paper develops a normal form for rational proper maps of balls using exhaustion functions, revealing invariants and classifications, especially for cubic maps, and explores polynomial equivalences.

Contribution

It introduces a normal form based on exhaustion functions, identifies spherical invariants, and classifies rational proper maps of balls, including cubic and polynomial cases.

Findings

Normal form for rational proper maps of balls established

Singular values of the quadratic denominator are invariants

Classification of cubic maps and polynomial equivalences provided

Abstract

We study a relationship between rational proper maps of balls in different dimensions and strongly plurisubharmonic exhaustion functions of the unit ball induced by such maps. Putting the unique critical point of this exhaustion function at the origin leads to a normal form for rational proper maps of balls. The normal form of the map, which is up to composition with unitaries, takes the origin to the origin, and it normalizes the denominator by eliminating the linear terms and diagonalizing the quadratic part. The singular values of the quadratic part of the denominator are spherical invariants of the map. When these singular values are positive and distinct, the normal form is determined up to a finite subgroup of the unitary group. We also study which denominators arise for cubic maps, and when we do not require taking the origin to the origin, which maps are equivalent to polynomials.