There are Salem numbers with trace $-3$ and every degree at least $34$

Giacomo Cherubini; Pavlo Yatsyna

There are Salem numbers with trace $-3$ and every degree at least $34$

Number Theory 2024-03-26 v2

Authors: Giacomo Cherubini , Pavlo Yatsyna

Abstract

We prove that there exist Salem numbers with trace $-3$ and every even degree $\geq 34$ . 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 $\leq 30$ , our result is optimal up to possibly the single value $32$ , for which it is expected there are no such numbers.

Keywords

salem numbers trace degree numerical search algebraic integers

Cite

@article{arxiv.2301.06536,
  title  = {There are Salem numbers with trace $-3$ and every degree at least $34$},
  author = {Giacomo Cherubini and Pavlo Yatsyna},
  journal= {arXiv preprint arXiv:2301.06536},
  year   = {2024}
}

Comments

7 pages; changed title; to appear in Exp. Math

There are Salem numbers with trace $-3$ and every degree at least $34$

Abstract

Keywords

Cite

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