Casas-Alvero Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K$ be any field of characteristic zero. For $f\in K[X]$, write $f^{(j)}$ for its $j$-th ordinary formal derivative, and let $\gcd(f,f^{(j)})$ be a greatest common divisor in $K[X]$. A nonconstant common factor means that this greatest common divisor has positive degree; the factor is allowed to depend on $j$. Write $c\ne0$ for the leading coefficient of $f$.

Statement recorded in the supplied source.

For every characteristic-zero field $K$ and every polynomial $f\in K[X]$ of degree $d\ge1$, does \[ \deg\gcd(f,f^{(j)})\ge1\quad\text{for every }1\le j\le d-1 \] imply that there is an $a\in K$ with $f(X)=c(X-a)^d$? Equivalently, after division by its leading coefficient, must $f$ be a $d$-th power of a monic linear polynomial? [S2]

Short English statement

If a polynomial shares a root with each of its nonconstant-order derivatives, must all its roots coincide? Different derivatives may share different roots a priori.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1107 (ALG-007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Soham Ghosh, A finiteness result towards the Casas--Alvero Conjecture, arXiv:2402.18717, revised version, Section 1, Conjecture CA. https://arxiv.org/abs/2402.18717.