The maximal conjugate of a non-reciprocal algebraic integer
For an algebraic integer \(a\) denote by \(\overline{|a|}\) the maximal modulus of conjugates of \(a\). Assume that \(a\) is non-reciprocal, i.e. its minimal polynomial \(P\) does not satisfy the identity \(P(x)=x^nP(1/x)\), with \(n\) denoting the degree of \(a\). \textit{J. W. S. Cassels} [J. Math. Sci. 1, 1-8 (1966; Zbl 0147.30803)] proved the inequality \(\overline{|a|}>1+0.1/n\), and from a later result of \textit{C. J. Smyth} [Bull. Lond. Math. Soc. 3, 169-175 (1971; Zbl 0235.12003)] the stronger inequality \(\overline{|a|}>1+c/n\) with \(c=0.2811\dots\) follows. The author shows that for \(n>n_0(\varepsilon)\) (with effective \(n_0(\varepsilon)\)) one can take \(c=\omega-\varepsilon\) where \(\omega=0.3096\dots\) is defined as a root of a rather complicated transcendental equation.
scientific article; zbMATH DE number 1131034