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.

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