Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022

Francesco Amoroso [a] and Sinnou David [b,c]

On the Galois group of lacunary polynomials

Pages: 19-29
Received: 3 May 2021
Accepted in revised form: 14 September 2021
Mathematics Subject Classification: 11G50, 11R06, 11R09, 11R27.
Keywords: Lehmer problem, heights, Galois groups, lacunary polynomials.
Authors address:
[a]: Laboratoire de mathématiques Nicolas Oresme, CNRS UMR 6139, Normandie Université, Université de Caen, Campus II, BP 5186, 14032 Caen Cedex, France
[b]: Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS UMR 7586, Sorbonne Université, 4, place Jussieu, 75005 Paris, France
[c]: CNRS UMI 2000 Relax, Institute of Mathematical Sciences, IV Cross Road, CIT Campus Taramani, Chennai 600 113, Tamil Nadu, India

A Roberto,con affetto.Francesco

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Abstract: We show that the Galois group defined by the roots of a lacunary polynomial is large in the sense that it grows faster than polynomially with the degree. Lacunary polynomials are the standard way of producing examples of algebraic numbers with small Weil height. One of the key tools in our proof is a relative lower bound by Delsinne for the height of a point in a power of the multiplicative group.

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