Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals → CMB
Abstract view

# On Cauchy--Liouville--Mirimanoff Polynomials

Published:2007-06-01
Printed: Jun 2007
• Pavlos Tzermias
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

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 $3 \leq d \leq 11$. 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 P\'{o}lya and Szeg\"{o}. These conditions are \emph{conjecturally} also necessary for irreducibility.
 MSC Classifications: 11G30 - Curves of arbitrary genus or genus \$ 11R09 - Polynomials (irreducibility, etc.) 12D05 - Polynomials: factorization 12E10 - Special polynomials

© Canadian Mathematical Society, 2014 : https://cms.math.ca/