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An inequality about irreducible factors of integer polynomials. (English) Zbl 0648.12002
Let \(F(X)=a_ 0+a_ 1X+a_ 2X^ 2+...+a_ dX^ d\) be a polynomial of degree d with complex coefficients. The author gives an upper bound for \(\max \{| F(z)|:| z| =1\}\) in the case when F(X) is an irreducible polynomial with integral coefficients. The upper bound involves the measure of F, namely, \(M(F)=| a_ d| \prod^{d}_{j=1} \max \{1,| z_ j|\},\) where \(z_ 1,...,z_ d\) are the complex roots of F. A special case of the author’s result is the following inequality: \((\sum^{d}_{i=0}| a_ i|^ 2)^{1/2}\leq e^{\sqrt{d}}(d+2\sqrt{d+2}^{1+\sqrt{d}}M(F)^{1+\sqrt{d}}\).
Reviewer: K.S.Williams

MSC:
11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
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