An inequality about irreducible factors of integer polynomials (Q1104972)

Let \(F(X)=a_0+a_1X+a_2X^2+ \ldots +a_dX^d\) 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 \(z_1,\ldots, z_d\) are the complex roots of \(F\). A special case of the author's result is the following inequality: \[ \left(\sum^d_{i=0}| a_i|^2\right)^{1/2}\le e^{\sqrt{d}}(d+2\sqrt{d}+2)^{1+\sqrt{d}}M(F)^{1+\sqrt{d}}. \]