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arXiv · 1211.2059

La conjecture de Casas Alvero pour les degrés $5p^{e}$

Mustapha Chellali
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Abstract

According to Casas Alvero conjecture, if a one variable polynomial of degree $n$ over a field of characteristic 0 is not prime with each of the $n-1$ first derivees, then it is of the form $c (X-r)^{n}$. Let $p$ be a prime number and an integer $e$, the conjecture is showed to be true for polynomials of degree $p^{e}, 2p^{e}, 3p^{e} (p neq 2) and 4p^{e} (p neq 3,5, 7) $. In this work we show that the conjecture is true for polynomials of degree $5p^{e} (p neq 2,3,7,11,131,193,599,3541,8009) $ . It also corrects an error in Draisma and Jong (2011) for the polynomials of degree $4p^{e}$ ----- Selon la conjecture de Casas Alvero, si un polynôme á une variable de degré $n$ sur un corps commutatif de caractéristique 0 est non premier avec chacune de ses $n-1$ premiéres dérivés, alors il est de forme $c(X-r)^{n}$. Soient $p$ un nombre premier et $e$ un entier, la conjecture a été démontrée pour les polynômes de degré $p^{e},2p^{e}, 3p^{e} (p\neq 2) et 4p^{e} (p\neq 3,5,7)$. Dans ce travail on montre que la conjecture est vrai pour les polynômes de degré $5p^{e} (p\neq 2,3,7,11,131,193,599,3541,8009)$. On corrige aussi une erreur dans Draisma et Jong (2011) pour les degré $4p^{e}$

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Mustapha Chellali. 2012-11-09. La conjecture de Casas Alvero pour les degrés $5p^{e}$. https://arxiv.org/abs/1211.2059

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