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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’skiĭ-type inequalities) 41A63 Multidimensional problems
##### Keywords:
estimates; polynomials; supnorms; degree
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