Totally positive algebraic integers with small trace

Abstract

The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for S k S_k , and reduce sharply the computing time. We are then able to push the computation to degree 15 15 and prove that there is no such totally positive algebraic integer with absolute trace 1.8 1.8 . As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to 1.793145 ⋯ 1.793145\cdots .

Potential energy of totally positive algebraic integers

“…Even stronger bounds than (1.14) can be obtained by the method of auxiliary polynomials introduced by Smyth [9], once we observe that the potential energy is the trace of the polynomial with roots (x i − x j ) 2 . The best constant in the trace problem has been obtained with Smyth's method and is 1.793145, due to Wang-Wu-Wu [10].…”

“…Their results have applications to what is nowadays called the Schur-Siegel-Smyth trace problem (see e.g. [1] for an overview of the problem and [8,10] for the current state of the art).…”

Algebraic integers with conjugates in a prescribed distribution

“…In his letter to Smyth dated 24 February 1998, Serre showed that, for any t > 0 and γ ∈ [0, 1], the inequality However, we know that this measure is not perfectly optimized for the trace problem. Indeed, the 14 totally positive algebraic integers α with tr(α)/ deg(α) at most 1.793 have all conjugates lying in the support of µ [22], while the support of an optimal measure would contain none of these points by Proposition 5.8.…”

“…It has long been known that there are infinitely many totally positive algebraic integers α with tr(α)/ deg(α) at most 2, with Siegel citing the family (1.1) 4 cos 2 (π/p) : p an odd prime as an example [17]. At the other end, the totally positive algebraic integers satisfying tr(α) < 1.793145 • deg(α) have been determined explictly; there are a total of 14 such algebraic integers [22]. This last result fits into the framework of the following problem, which was codified by Borwein in [3].…”

“…Smyth's original article gives an instance of (1.3) with λ = 1.7719 and about 15 auxiliary polynomials. The current state of the art [22] gives an instance with λ = 1.793145 and 130 auxiliary polynomials.…”