From the text: Suppose that \(P(x)=x^d+a_1x^{d-1}+\ldots+a_d\) is a polynomial with integer coefficients, irreducible, and with all roots real and positive. In a remarkable paper of 1918, I. Schur [Math. Z. 1, 377–402 (1918; JFM 46.0128.03)] proved that if \(c<\sqrt e\), then there are only finitely many such polynomials for which the average of the roots, equal to \(-a_1/d\), is less than \(c\). The Schur-Siegel-Smyth trace problem asks for the largest value of \(c\) for which the same conclusion holds. In this paper we give an account of the history of the problem, the latest results, and its relations with other problems in number theory.
The paper is divided into five sections and two appendices. After the introductory first section, Section 2 is devoted to the work of I. Schur, C. L. Siegel and C. J. Smyth on the trace problem. In Section 3 we explain the method of auxiliary functions and give the best results known. Section 4 deals with the relation between the trace problem and the integer Chebyshev problem, and Section 5 is dedicated to the special case of cyclotomic algebraic integers.
Appendix A gives the bets result, as far as we know, for the trace problem. Appendix B contains a letter from J.-P. Serre to C. Smyth.
For the entire collection see [Zbl 1139.11002].