An operator algebraic approach for generalized Cardano polynomials

Abstract

We develop an operator algebraic framework for generalized Cardano polynomials and show how their structure naturally leads to an operator formulation of Cardano method that is compatible with tools and concepts from quantum information theory. The generalized Cardano polynomials are constructed as a generalization of classical theory of Cardano formula for cubic equation, as well as through the spectral properties of the circular operator, that embeds Cardano type identities into their spectral theory. The construction clarifies the algebraic structure and solvability of a family of two parameters odd order polynomials, classically and through operator methods familiar in QIT, including Fourier transforms and spectral calculus on operator algebras. As applications, we show connections to Cebyshev polynomials and the solution of the quartic order Ferrari equation.