Complexity results for CR mappings between spheres

TL;DR

This paper investigates the complexity of CR mappings between spheres, revealing infinite degrees where uniqueness fails and showing the gap phenomenon does not occur beyond certain dimensions, using number theory and explicit computations.

Contribution

It introduces new results on the degrees of CR mappings, employing elementary number theory, Pell equations, and the postage stamp problem to analyze their properties.

Findings

Infinitely many degrees where sharp examples are not unique

Gap phenomenon for proper mappings ceases beyond certain dimensions

Uses Pell equations and postage stamp problem in proofs

Abstract

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.