A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero 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 $n$, 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 $n{p}^{\ell }$, $\ell \in \mathbb{N}$, 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 $\left(\genfrac{}{}{0ex}{}{\frac{n^2-n}{2}}{n-2}\right)!\prod _{i=1}^{n-1}{\left(\genfrac{}{}{0ex}{}{i+n-2}{n-2}\right)}^{\left(\genfrac{}{}{0ex}{}{d-i+n-2}{n-2}\right)}$.

arXiv admin note: text overlap with arXiv:2307.05997