Uniqueness of certain polynomials constant on a line

TL;DR

This paper investigates polynomials with positive coefficients that are constant on a line, characterizes those achieving the maximum degree for a given number of terms, and classifies them up to degree 17, with implications for CR map classification.

Contribution

It provides new insights into the structure of sharp polynomials and offers a complete classification up to degree 17 using theoretical and computational methods.

Findings

Proved new results about the form of sharp polynomials.

Established a complete classification of these polynomials up to degree 17.

Confirmed the sharp degree bound for polynomials with positive coefficients.

Abstract

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$ when $x+y=1$. A sharp estimate $d \leq 2N-3$ is known. In this paper we study the $p$ for which equality holds. We prove some new results about the form of these "sharp" polynomials. Using these new results and using two independent computational methods we give a complete classification of these polynomials up to $d=17$. The question is motivated by the problem of classification of CR maps between spheres in different dimensions.