具有负迹的Salem数的最小次数
The Minimal Degree of Salem Numbers with Negative Trace
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中文总结 AI 辅助
该研究改进了Salem数最小次数的上界，证明其最小次数与迹的绝对值的平方相关，且对每个指定负迹，超过某偶次时均存在对应Salem数。
中文摘要 AI 辅助
我们证明，具有指定负迹的Salem数的最小次数，在显式常数和低阶项的范围内，下界为$T^2/\log T$，上界为$T^2\log T$，其中$T$是迹的绝对值。这改进了McKee和Smyth之前的双指数上界。基于他们的构造，我们还证明，对于每个指定的负迹，存在常数$d_0$，使得在超过$d_0$的每个偶次中都存在该迹的Salem数。
英文摘要
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 $d_0$.
