Polynomials constant on a hyperplane and CR maps of spheres

TL;DR

This paper establishes a precise degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative monomials, confirming a conjecture and characterizing extremal cases across all dimensions.

Contribution

It proves the conjectured degree bound in all dimensions and characterizes all extremal polynomials achieving this bound.

Findings

Proved the sharp degree bound for polynomials on hyperplanes in all dimensions.

Provided a complete description of extremal polynomials in dimensions 4 and higher.

Confirmed the bound's implications for monomial CR mappings of spheres.

Abstract

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.