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Polynomials in many variables: Real vs complex norms. (English) Zbl 0827.32001
The purpose of this present paper is to study polynomials \(P\) in many variables \(x_1, x_2, \ldots, x_N\), of total degree \(n\), written in the form \[ P(x_1, \ldots, x_N) = \sum_{|\alpha |\leq n} a_\alpha x_1^{\alpha_1} x_2^{\alpha_2} \cdots x^{\alpha_N}_N, \] with \(\alpha = (\alpha_1, \ldots, \alpha_N)\) and \(|\alpha |= \alpha_1 + \cdots + \alpha_N\).
Among other results the authors prove two central problems in connection to the supnorms of the polynomials \(P\), i.e. on the one side the supnorms in the real case and on the other side in the complex case. Therefore they give for example the very interesting estimate \[ |P |_{\infty, \mathbb{R}} \geq {1 \over \bigl( 2 + \sqrt 2 \bigr)^n + \bigl( 2 - \sqrt 2 \bigr)^n} |P |_{\infty, \mathbb{C}}, \] which shows the connection between the real \(|\cdot |_{\infty, \mathbb{R}}\)-norm and the complex supnorm \(|\cdot |_{\infty, \mathbb{C}}\).
One main problem is the question, whether it is enough to know the quantity of the first leading coefficient \(\Sigma^N_1 |a_l |\) of \(x^n_l\) with no information at all on the rest of \(P\), and to give bounds for its supnorms. Useful are some ideas in the area of Partial Differential Equations (reducing the polynomial \(P)\).
Further the authors consider the case when the polynomial \(P\) has a given degree, or some concentration at a given degree.

MSC:
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
26C05 Real polynomials: analytic properties, etc.
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A63 Multidimensional problems
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