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

TL;DR

This paper characterizes and bounds the set of primes where the Casas-Alvero conjecture fails for polynomials of a given degree, providing explicit bounds based on combinatorial formulas.

Contribution

It offers an explicit description and upper bound for the set of bad primes in any degree, advancing understanding of the conjecture's validity over different characteristics.

Findings

Explicit description of bad primes for any degree n

An upper bound on bad primes involving combinatorial expressions

Implications for proving the conjecture in characteristic zero

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 $np^\ell$, $\ell\in \mathbb{N}$, 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 $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.