On Cauchy–Liouville–Mirimanoff Polynomials

Tzermias, Pavlos

Canadian mathematical bulletin, Tome 50 (2007) no. 2, pp. 313-320

Résumé

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.

Détail

DOI : 10.4153/CMB-2007-030-7

Mots-clés : 11G30, 11R09, 12D05, 12E10

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