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On Cauchy–Liouville–Mirimanoff Polynomials

Published online by Cambridge University Press:  20 November 2018

Pavlos Tzermias
Pavlos Tzermias*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, U.S.A. e-mail: tzermias@math.utk.edu
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Abstract

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Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy–Liouville–Mirimanoff polynomials to show that the intersection of the Fermat curve of degree $p$ with the line $X+Y=Z$ in the projective plane contains no algebraic points of degree $d$ with $3\le d\le 11$. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of Pólya and Szegö. These conditions are conjecturally also necessary for irreducibility.

Keywords

11G3011R0912D0512E10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 2 , 01 June 2007 , pp. 313 - 320
Copyright
Copyright © Canadian Mathematical Society 2007

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