Lebl, Jiří; Peters, Han Polynomials constant on a hyperplane and CR maps of spheres. (English) Zbl 1278.14073 Ill. J. Math. 56, No. 1, 155-175 (2012). The authors prove the following result. Let \(n\geq2\) and let \(p\) be a polynomial with nonnegative coefficients of degree \(d\) in \(n\) dimensions such that \(p(x_1,\dots,x_n)=1\) whenever \(x_1+\dots+x_n=1\). Moreover, let \(N(p)\) be the number of nonzero coefficients of a polynomial \(p\). Then, for \(n=2\), \[ d\leq2N(p)-3 \] and, for \(n>2\), \[ d\leq\frac{N(p)-1}{n-1}. \] Moreover, when \(n=2\), the equality in the above formula can hold for any odd degree \(d\) and the right hand side can be exactly one larger than \(d\) for any even degree \(d\). When \(n>2\), then for every \(d\) there exist a polynomial \(p\) such that the equality in the above formula holds.This theorem generalizes the results obtained by J. P. D’Angelo, Š. Kos and E. Riehl in [J. Geom. Anal. 13, No. 4, 581–593 (2003; Zbl 1052.26016)] (case \(n=2\)) and by the authors in [Mosc. Math. J. 11, No. 2, 285–315 (2011; Zbl 1258.14068)] (case \(n=3\)). It also solves the problem posed by D’Angelo, who conjectured that inequality \[ d\leq\frac{N(p)-1}{n-1} \] holds for every \(n>2\). Reviewer: Pawel Zapalowski (Kraków) Cited in 12 Documents MSC: 14P99 Real algebraic and real-analytic geometry 05A20 Combinatorial inequalities 32H35 Proper holomorphic mappings, finiteness theorems 11C08 Polynomials in number theory Keywords:polynomials constant on a hyperplane; CR mappings of spheres; degree estimates; Whitney polynomials; Newton diagrams Citations:Zbl 1052.26016; Zbl 1258.14068 PDFBibTeX XMLCite \textit{J. Lebl} and \textit{H. Peters}, Ill. J. Math. 56, No. 1, 155--175 (2012; Zbl 1278.14073) Full Text: arXiv Euclid