Uniqueness of certain polynomials constant on a line

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⩽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.