Finding All Salem Numbers of Trace $-1$ and Degree up to $20$

Abstract

In 1999, C. J. Smyth proved that, for all $d \geq 4$, there are Salem numbers of degree $2d$ and trace $-1$, and that the number of them is greater than $0.1387d/(\log \log d)^2$. He gave also all Salem numbers of trace $-1$ and degree $2d = 8,10,12,14$. In this paper, we complete the table of the minimal polynomials of all Salem numbers of trace $-1$ and degree $2d = 16, 18, 20$, and we conjecture a new lower bound of the numbers of such Salem numbers.