## 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 $$n>2$$.

### MSC:

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

### Citations:

Zbl 1052.26016; Zbl 1258.14068
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