An operator algebraic approach for generalized Cardano polynomials

TL;DR

This paper introduces an operator algebraic framework for generalized Cardano polynomials, linking classical solutions of cubic equations with quantum information tools, spectral theory, and polynomial identities.

Contribution

It develops a novel operator algebraic approach to generalized Cardano polynomials, integrating spectral theory and quantum information concepts for solving polynomial equations.

Findings

Operator algebraic formulation of Cardano polynomials

Connection to spectral properties of circular operators

Applications to quartic and Chebyshev 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.