The Minimal Degree of Salem Numbers with Negative Trace

Abstract

We show that the minimal degree of a Salem number with prescribed negative trace is bounded below by $T^2/\log T$ and above by $T^2\log T$, up to explicit constants and lower-order terms, where $T$ is the absolute value of the trace. This improves the previous doubly exponential upper bound of McKee and Smyth. Building on their construction, we also prove that for every prescribed negative trace there exists a constant $d_0$ such that Salem numbers of that trace exist in every even degree exceeding $d_0$.