## Salem numbers with minimal trace

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Qiong Chen and Qiang Wu
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## Abstract:

In this paper, a new method to compute lower and upper bounds for Salem numbers with a given trace and a given degree is given. With this method, it is proven that the smallest trace of Salem numbers of degree $22$ is $-1$. Further, new lower bounds for degree of Salem numbers with minimal trace $-5$ and $-6$ are given. All Salem numbers of trace $-2$ and degree $24$, $26$ are given. This includes $7$ additional Salem numbers of degree $26$ beyond what was previously known. The auxiliary functions related to Chebyshev polynomials, which are adapted to Salem number, are used in this work.## References

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## Additional Information

**Qiong Chen**- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, People’s Republic of China
- Email: 3418484955@qq.com
**Qiang Wu**- Affiliation: Department of Mathematics, Southwest University of China, 2 Tiansheng Road Beibei, 400715 Chongqing, People’s Republic of China
- Email: qiangwu@swu.edu.cn
- Received by editor(s): July 13, 2022
- Received by editor(s) in revised form: October 17, 2022, November 3, 2022, and January 2, 2023
- Published electronically: March 7, 2023
- Additional Notes: This work was supported by the Natural Science Foundation of China (Grant numbers: 12071375, 12271442).

The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1779-1790 - MSC (2020): Primary 11C08, 11R06, 11Y40
- DOI: https://doi.org/10.1090/mcom/3833
- MathSciNet review: 4570341