Polynomials constant on a hyperplane and CR maps of hyperquadrics

TL;DR

This paper establishes sharp degree bounds for polynomials constant on hyperplanes and explores their connection to CR maps of hyperquadrics, with implications for conjectures on degree bounds in complex geometry.

Contribution

It provides new sharp degree bounds for polynomials on hyperplanes and extends these results to monomial CR maps of hyperquadrics, especially spheres.

Findings

Sharp degree bounds for polynomials in 2 and 3 dimensions.

Extension of bounds to monomial CR maps of hyperquadrics.

Support for conjectures on degree bounds in complex geometry.

Abstract

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support generalizing a conjecture on the degree bounds to the more general case of hyperquadrics.