C*-algebraic Casas-Alvero Conjecture

Abstract: Based on Casas-Alvero conjecture \textit{[J. Algebra, 2001]} we formulate the following conjecture.\ \textbf{C*-algebraic Casas-Alvero Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra, $n\in \mathbb{N}$ and let $P(z) = (z-a_1)(z-a_2)\cdots (z-a_n)$ be a polynomial over $\mathcal{A}$ with $a_1, a_2, \dots, a_n \in \mathcal{A}$. If $P$ shares a common zero with each of its (first) $n-1$ derivatives, then it is $n^\text{th}$ power of a linear monic C*-algebraic polynomial.}\ We show that C*-algebraic Casas-Alvero Conjecture holds for C*-algebraic polynomials of degree 2.