C*-algebraic Casas-Alvero Conjecture

Based on Casas-Alvero conjecture $\mathit{\text{[J.}}\mathit{\text{Algebra,}}\mathit{\text{2001]}}$ we formulate the following conjecture.\ $\mathbf{\text{C*-algebraic}}\mathbf{\text{Casas-Alvero}}\mathbf{\text{Conjecture}}\mathbf{\text{:}}\mathbf{\text{Let}}\mathcal{A}\mathbf{\text{be}}\mathbf{\text{a}}\mathbf{\text{commutative}}\mathbf{\text{C*-algebra,}}n\in \mathbb{N}\mathbf{\text{and}}\mathbf{\text{let}}P(z)=(z-a_1)(z-a_2)\cdots (z-a_n)\mathbf{\text{be}}\mathbf{\text{a}}\mathbf{\text{polynomial}}\mathbf{\text{over}}\mathcal{A}\mathbf{\text{with}}{a}_1,a_2,\dots ,a_n\in \mathcal{A}\mathbf{\text{.}}\mathbf{\text{If}}P\mathbf{\text{shares}}\mathbf{\text{a}}\mathbf{\text{common}}\mathbf{\text{zero}}\mathbf{\text{with}}\mathbf{\text{each}}\mathbf{\text{of}}\mathbf{\text{its}}\mathbf{\text{(first)}}n-1\mathbf{\text{derivatives,}}\mathbf{\text{then}}\mathbf{\text{it}}\mathbf{\text{is}}{n}^{\mathbf{\text{th}}}\mathbf{\text{power}}\mathbf{\text{of}}\mathbf{\text{a}}\mathbf{\text{linear}}\mathbf{\text{monic}}\mathbf{\text{C*-algebraic}}\mathbf{\text{polynomial.}}$\ We show that C*-algebraic Casas-Alvero Conjecture holds for C*-algebraic polynomials of degree 2.