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Estimates for the uniform norm of complex polynomials in the unit disk

✍ Scribed by Richard Fournier; Gérard Letac; Stephan Ruscheweyh

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
106 KB
Volume
283
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let ‖ · ‖ denote the uniform norm in the unit disk of the complex plane ℂ. The main result in this note is as follows: For any complex polynomial P of degree at most n and any α ∈ ℂ the inequality
‖P ‖ ⩽ (n + 1)(‖z P (z) + α ‖ ‐ |α |)
holds. For any α ≠ 0 the factor n + 1 is best possible, and we determine the cases of equality (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Bounds for the Height of a Factor of a P
Bounds for the Height of a Factor of a Polynomial in Terms of Bombieri's Norms: II. The Smallest Factor
✍ David W. Boyd 📂 Article 📅 1993 🏛 Elsevier Science 🌐 English ⚖ 436 KB

Let \(f(x)\) be a polynomial of degree \(n\) with complex coefficients, which factors as \(f(x)=\) \(g(x) h(x)\). Let \(H(g)\) be the maximum of the absolute value of the coefficients of \(g\). For \(1 \leq p \leq \infty\), let \([f]_{p}\) denote the \(p^{\text {th }}\) Bombieri norm of \(f\). This