On the length of totally positive algebraic integers
For a polynomial \[ P(X)= X^d+ a_{d-1} X^{d-1}+ \cdots+ a_0= \prod_{k=1}^d (X- \alpha) \] put \(R_1 (P)= \prod_{k=1}^d (1+|\alpha_k |)\) and \(R(P)= R_1 (P)^{1/d}\). The author studies the set \({\mathcal R}\) of the values of \(R(P)\), where \(P\) runs over irreducible polynomials \(\neq X, X-1\) with integral coefficients having all roots positive. He determines the five smallest numbers in \({\mathcal R}\) (which correspond to certain polynomials considered by \textit{C. J. Smyth} [Math. Comput. 37, 205-208 (1981; Zbl 0475.12001)]) and shows that this set is dense in a certain halfline \([l,\infty ]\) with \(l= 2.3768 \dots\;\).
scientific article; zbMATH DE number 809107