There are Salem Numbers with Trace –3 and Every Degree At Least 34

Abstract

We prove that there exist Salem numbers with trace –3 and every even degree $\ge 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace –3 and degree $\le 30$, our result is optimal up to possibly the single value 32, for which it is expected there are no such numbers.

KEYWORDS:

• Salem number
• trace
• polynomial
• interlacing
• cyclotomic

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 11R06
• 11Y40
• 11R09
• 11C08

Acknowledgements

The authors would like to thank the referees for their valuable comments.

Additional information

Funding

G.C. was supported by Czech Science Foundation GACR, grant 21-00420M. P.Y. was supported by Charles University, projects PRIMUS/24/SCI/010 and UNCE/24/SCI/022, the Academy of Finland (grants 336005 and 351271, PI C. Hollanti), and by MATINE, Ministry of Defence of Finland (grant 2500M-0147, PI C. Hollanti).