Let \(\mathrm{Tr}(\alpha)\) be the trace of an algebraic number, \(\alpha\), of degree \(d\) and \(\mathrm{tr}(\alpha)=\mathrm{Tr}(\alpha)/d\), be its absolute trace. A well-known problem, known as the Schur-Siegel-Smyth trace problem, is formulated as follows: given a real number \(\rho<2\), show that all but finitely many totally positive algebraic integers \(\alpha\) have \(\mathrm{tr}(\alpha) > \rho\). A. Smith recently proved the surprising result that this is not true [“Algebraic integers with conjugates in a prescribed distribution”, Preprint, arXiv:2111.12660]. He showed that there are infinitely many totally positive algebraic integers \(\alpha\) with \(\mathrm{tr}(\alpha)<1.8984\).
One can pose the same problem for other measures too. That is the subject of the paper under review. Letting \(\alpha_{1}=\alpha,\ldots,\alpha_{d}\) be the algebraic conjugates of \(\alpha\), the author defines the \(G\)-measure, \(G(\alpha)=\sum_{i=1}^{d} \left( \left| \alpha_{i} \right| + 1/\left| \alpha_{i} \right| \right)\), and the absolute \(G\)-measure, \(g(\alpha)=G(\alpha)/d\). She proves that a family of examples due to Smyth shows that \(4\) is a limit point of the set of all \(g(\alpha)\) for totally positive algebraic integers \(\alpha\) and finds a lower bound for all limit points. More precisely, Theorem 1.2 shows that unless \(\alpha\) is one of five values, then \(g(\alpha) \geq 3.024561\) for all totally positive algebraic integers.
It would be interesting to see how small the gap between the upper bound and lower bound can be reduced and also if a Smith-type result holds for the absolute \(G\)-measure (and other measures).
One can also consider what happens for algebraic integers that are not totally positive. Here it is natural to look at what happens in sectors. That is to find the minimum value of \(g(\alpha)\) when all the conjugates of \(\alpha\) are in the sector \(S_{\theta}=\left\{ z \in \mathbb{C}: |\mathrm{arg} (z)| \leq \theta \right\}\) for \(0<\theta<\pi/2\) (the study of totally positive algebraic integers corresponds to studying \(S_{0}\)). One of the things that makes such a question interesting is a conjecture of G. Rhin and C. Smyth [Math. Comp. 64, 295–304 (1995; Zbl 0820.11064)] regarding the behaviour of \(M(\alpha)^{1/d}\) in such sectors where \(M(\alpha)\) is the Mahler measure of \(\alpha\). They conjectured that if \(M(\alpha)^{1/d} \geq c(\theta)\) when all the conjugates of \(\alpha\) lie in \(S_{\theta}\), then \(c(\theta)\) is a “staircase” decreasing function of \(\theta\), which is constant except for finitely many left discontinuities in any closed subinterval of \([0,\pi)\). We have some results on some portions of the stairs for \(M(\alpha)^{1/d}\), but there are still many gaps.
Here the author finds all the complete and consecutive stairs in such a staircase for \(g(\alpha)\), along with the exact values of \(g(\alpha)\) on each such stair, for \(0 \leq \theta \leq 86.24°\). These stairs and information about them are given in Table 1. This confirms a version of the Rhin-Smyth conjecture for \(g(\alpha)\) except for a small sector of \([0,90°)\). It is the first time that this conjecture has been verified for such a sizeable interval.
The proof is computational and uses the method of explicit auxiliary functions that the author has used effectively in the past for other height-related problems. It also includes a refinement of this method due to the author and G. Rhin [Math. Comput. 84, No. 296, 2927–2938 (2015; Zbl 1325.11108)] which uses induction to obtain the polynomials used in the construction of these auxiliary functions.