A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture - Centre Henri Lebesgue
Pré-Publication, Document De Travail Année : 2023

A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture

Résumé

The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $n$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $n$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $np^\ell$, $\ell\in\N$, where $p$ is a good prime for $n$. In this paper we give an explicit description of the set of bad primes in any given degree $n$. In particular, we show that if the conjecture holds in degree $n$ then the bad primes for $n$ are bounded above by $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1} \binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.
Fichier principal
Vignette du fichier
casas-badprimes.pdf (95.34 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04791489 , version 1 (20-11-2024)

Identifiants

  • HAL Id : hal-04791489 , version 1

Citer

Daniel Schaub, Mark Spivakovsky. A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture. 2024. ⟨hal-04791489⟩
0 Consultations
0 Téléchargements

Partager

More