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Abstract
We show that the derivative of the minimal polynomial of a Salem (resp. Pisot) number
α of degree
d
(resp. d
≥ 2 ) has
d
− 2 zeros with
modulus less than
1
and a real zero
𝜃
> 1
satisfying
| 𝜃
−
( d
− 1 ) α ∕ d |
< 1 ∕ d
(resp. | 𝜃
−
( d
− 1 ) α ∕ d |
< 1 ∕ d ,
except when
α
belongs to a set of nine explicitly listed elements).
Keywords
Salem polynomials, Pisot polynomials, Gauss–Lucas theorem,
Cohn's theorem
Mathematical Subject Classification
Primary: 11R06, 12D10
Secondary: 11C08, 30C15
Milestones
Received: 10 May 2021
Revised: 18 January 2022
Accepted: 1 February 2022
Published: 15 October 2022