Salem numbers of negative trace
Chris J. Smyth · Mathematics of Computation · 1999
We prove that, for all d ≥ 4 d\geq 4 , there are Salem numbers of degree 2 d 2d and trace − 1 -1 , and that the number of such Salem numbers is ≫ d / ( log log d ) 2 \gg d/\left ( \log \log d\right ) ^{2} . As a consequence, it follows that the number of totally positive algebraic integers of degree d d and trace 2 d − 1 2d-1 is also ≫ d / ( log log d ) 2 \gg d/\left ( \log \log d\right ) ^{2} .