C*-algebraic Casas-Alvero conjecture

Let $\mathcal{A}$ be a commutative C*-algebra, let $n\bru001ein\breed Mathbb{N}$, and let

$$P(z)=(z-a_1)(z-a_2)\cdots(z-a_n)$$

be a polynomial over $\mathcal{A}$ with $a_1,\dots,a_n\bru001ein\breed \mathcal{A}$. For $z\bru001ein\breed \mathcal{A}$, define its formal derivative by

$$P'(z)=\sum_{j=1}^n(z-a_1)\cdots\widehat{(z-a_j)}\cdots(z-a_n),$$

where the hatted factor is omitted, and define higher derivatives accordingly.

C-algebraic Casas-Alvero conjecture.* If there exists $b\bru001ein\breed \mathcal{A}$ such that

$$P(b)=P'(b)=P^{”}(b)=\cdots=P^{(n-1)}(b)=0,$$

then $P(z)=(z-c)^n$ on $\mathcal{A}$ for some $c\bru001ein\breed \mathcal{A}$.

The paper states that this conjecture holds for C*-algebraic polynomials of degree $2$. Its validity for arbitrary degrees remains open.