A note on the Casas-Alvero Conjecture
Daniel Schaub, Mark Spivakovsky
TL;DR
This work analyzes the Casas–Alvero conjecture for a monic univariate polynomial of degree $d$ over a field of characteristic zero by encoding potential counterexamples through the system of resultants $R_i=\operatorname{Res}(f,H_i(f))$ with $H_i(f)$ the $i$-th Hasse derivative, linking the conjecture to the geometry of the variety defined by $R_1,...,R_{d-1}$. It furnishes a partial independence result: for indices $i\in\{d-3,d-2,d-1\}$, $R_i \notin \sqrt{(R_1,...,\breve{R_i},...,R_{d-1})}$, i.e., these $R_i$ cannot be generated up to radical by the others. The proof leverages real-rootedness properties of $f$ and its derivatives, the almost-counterexample construction of Draisma–de Jong and results of CLO, to derive a contradiction if such radical containment held. The authors also note a subsequent complete proof of the Casas–Alvero conjecture by Soham Ghosh in two preprints.
Abstract
The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let $f=x^d+a\_1x^{d-1}+\cdots+a\_1x \in K[a\_1,\ldots,a\_{d-1}][x]$ and let $R\_i = Res(f,H\_i(f))\in K[a\_1,\ldots,a\_{d-1}]$ be the resultant of $f$ and $H\_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\_1,\ldots,R\_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R\_1,\ldots,R\_{d-1})=d-1$ in $K[a\_1,\ldots,a\_{d-1}]$. In this paper we prove a very partial result in this direction : if $i \in \{d-3,d-2,d-1\}$ then $R\_i \notin \sqrt{(R\_1,\ldots,\breve{R\_i},\ldots,R\_{d-1}}$.