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 ⩽ 2 N - 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.
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