On the set of bad primes in the study of Casas-Alvero Conjecture

TL;DR

This paper investigates the set of primes for which the Casas-Alvero conjecture fails in specific degrees, by analyzing resultants and identifying bad primes that hinder the conjecture's proof.

Contribution

It computes distinguished monomials in resultants to identify bad primes for all degrees, advancing the understanding of the conjecture's prime-related obstructions.

Findings

List of bad primes for each degree d obtained

Identification of key monomials in resultants

Partial characterization of primes affecting the conjecture

Abstract

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 $d$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $d$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $dp^\ell$, $\ell\in\mathbb{N}$, where $p$ is a good prime for $d$. In this paper we calculate certain distinguished monomials appearing in the resultant $R(f,H_i(f))$ and obtain a (non-exhaustive) list of bad primes for every degree $d\in\mathbb{N}\setminus\{0\}$.