  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.
