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On the zeros of the derivatives of certain polynomials

Toufik Zaïmi

Vol. 11 (2022), No. 3, 205–214
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)αd| < 1d (resp. |𝜃 (d 1)αd| < 1d, 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
Authors
Toufik Zaïmi
Department of Mathematics and Statistics
Imam Mohammad Ibn Saud Islamic University (IMSIU)
Riyadh
Saudi Arabia