La conjecture de Casas Alvero pour les degr\'es $5p^{e}$

TL;DR

This paper proves the Casas Alvero conjecture for polynomials of degree 5p^e with specific primes p and corrects an earlier error related to degree 4p^e, advancing understanding of polynomial structure in characteristic zero.

Contribution

It extends the Casas Alvero conjecture verification to degree 5p^e for certain primes p and rectifies previous inaccuracies in related cases.

Findings

Confirmed the conjecture for degree 5p^e with specified primes p.

Corrected an error in prior work on degree 4p^e.

Expanded the class of polynomial degrees for which the conjecture holds.

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 polyn\^{o}me \'{a} une variable de degr\'{e} $n$ sur un corps commutatif de caract\'{e}ristique 0 est non premier avec chacune de ses $n-1$ premi\'{e}res d\'{e}riv\'{e}s, alors il est de forme $c(X-r)^{n}$. Soient $p$ un nombre premier et $e$ un entier, la conjecture a \'{e}t\'{e} d\'{e}montr\'{e}e pour les polyn\^{o}mes de degr\'{e} $p^{e},2p^{e}, 3p^{e} (p\neq 2) et 4p^{e} (p\neq 3,5,7)$. Dans ce travail on montre que la conjecture est vrai pour les polyn\^{o}mes de degr\'{e} $5p^{e} (p\neq 2,3,7,11,131,193,599,3541,8009)$. On corrige aussi une erreur dans Draisma et Jong (2011) pour les degr\'{e} $4p^{e}$