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On the absolute trace of polynomials having all zeros in a sector. (English) Zbl 1182.11050

The author finds a lower bound on the value of \(\text{Trace}(\alpha)/\deg(\alpha)\) for algebraic integers \(\alpha\) all of whose conjugates lie in a sector \(S_{\theta}=\{z \in \mathbb{C}: |\arg(z)|\leq \theta\}\) with \(0<\theta<\pi/2\). This bound is shown to be best possible for seven subintervals of \([0,\pi/2)\). The proof is computational. It is based on Smyth’s method of auxiliary functions who first applied it to \(\theta=0\) in 1981.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11Y40 Algebraic number theory computations
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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