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Polynomials constant on a hyperplane and CR maps of hyperquadrics. (English) Zbl 1258.14068

Let \(p\) be real polynomial of degree \(d\) in the variables \(x=(x_1,x_2,\dots,x_n)\) such that \(p(x)=1\) when \(s(x):=\sum_{j=1}^nx_j\). Let \(N(p)\) denote the number of distinct monomials of \(p\). In general, there is no function \(C\) such that \[ d\leq C(n,N(p)). \] Nevertheless, under the additional assumption on the positivity of all coefficients of \(p\) (a condition that arises naturally in CR geometry), J. P. D’Angelo, Š. Kos and E. Riehl [J. Geom. Anal. 13, No. 4, 581–593 (2003; Zbl 1052.26016)] proved that for \(n=2\) we have the following sharp estimate \[ d\leq 2N(p)-3. \] J. P. D’Angelo conjectured, that for polynomials with positive coefficients and \(n\geq3\) the best possible bound is \[ C(n,N(p))=\frac{N(p)-1}{n-1}. \] J. P. D’Angelo, J. Lebl and H. Peters [Mich. Math. J. 55, No. 3, 693–713 (2007; Zbl 1165.32005)] proved that
(a) if \(n\geq3\) then \[ d\leq\frac43\frac{2N(p)-3}{2n-3}; \]
(b) if \(n\) is sufficiently larger than \(d\) then we have the sharp bound \[ d\leq\frac{N(p)-1}{n-1}. \]
In the paper under review the authors extend the results above by proving that the D’Angelo conjecture holds for \(n=3\) and obtaining similar estimastes under weaker assumption of indecomposability (instead of positivity of coefficients). They prove the following:
1. if \(n=2\) and \(p\) is indecomposable then we have the sharp estimate \[ d\leq2N(p)-3; \]
2. if \(n=3\) and all coefficients of \(p\) are positive then we have the sharp estimate \[ d\leq\frac{N(p)-1}{2}; \]
3. if \(n=3\), \(p\) is indecomposable, and \(p\) satisfies additional mild assumption of no overhang then we have the sharp estimate \[ d\leq\frac{N(p)-1}{2}; \]
4. if \(n\geq2\) and \(p\) is indecomposable then \[ d\leq\frac43\frac{2N(p)-3}{2n-3}. \]
The authors also study the connection with monomial CR maps of hyperquadrics and prove in that case the similar bounds.

MSC:

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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