On some application of the spectral properties of the matrices
In this paper the authors prove that if \(\alpha\) is an algebraic integer of degree \(n\) which is not a root of unity and has norm different from \(\pm 1\) then it has a conjugate \(\alpha'\) whose modulus is greater than or equal to \(1+(\log 2)/n\). This result (Theorem 1) is related to a still open conjecture of Schinzel and Zassenhaus and is very well-known. The proof is using an inequality for the eigenvalues of a matrix that is composed with coefficients of the minimal polynomial of \(\alpha\). The statement of the other result (Theorem 2) asserting that the same holds for \(\alpha\) of norm \(\pm 1\) under the inequality \(n \varepsilon > (1-\varepsilon) \log 2\) is not clear, since \(\varepsilon\) is not defined.
scientific article; zbMATH DE number 6077464