On a Lower-Bound for the Absolute Value of a Polynomial of Several Complex Variables

✍ Scribed by B. Paneah

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 268 KB
• Volume: 78
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1994.1086

✦ Synopsis

For an arbitrary polynomial (P\left(z_{1}, z_{2}, \ldots, z_{n}\right)) in complex space (\mathbb{C}^{n}) we describe a set of nonnegative multi-indices (\alpha=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right)) such that for any (n)-tuple (\delta=\left(\delta_{1}, \delta_{2}, \ldots, \delta_{n}\right) \geq 0) (where (\delta_{j}=0) if (\alpha_{j}=0) ), one can find a system of "thin" sets (M_{j}) of widths (\leq \delta_{j}) in directions of the axes (z_{j}), respectively, (1 \leq j \leq n), for which outside their union the absolute value of a polynomial is bounded away from zero by ((\delta / \alpha)^{\alpha} \Gamma_{\alpha}\left(\Gamma_{\alpha}\right.) depends on (P) but not on (\delta) ). The prototype of this result is the well known Cartan's Theorem on a lower bound for the modulus of a polynomial (P(z), z \in \mathbb{C}^{1} . \quad \mathbb{C} 1994) Academic Press, inc.