Polynomials constant on a hyperplane and CR maps of spheres. (English) Zbl 1278.14073

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 \(n>2\).


14P99 Real algebraic and real-analytic geometry
05A20 Combinatorial inequalities
32H35 Proper holomorphic mappings, finiteness theorems
11C08 Polynomials in number theory
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