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On Cauchy--Liouville--Mirimanoff Polynomials

  Published:2007-06-01
 Printed: Jun 2007
  • Pavlos Tzermias
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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, 11R09, 12D05, 12E10 show english descriptions Curves of arbitrary genus or genus $
Polynomials (irreducibility, etc.)
Polynomials: factorization
Special polynomials
11G30 - Curves of arbitrary genus or genus $
11R09 - Polynomials (irreducibility, etc.)
12D05 - Polynomials: factorization
12E10 - Special polynomials
 

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