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An operator algebraic approach for generalized Cardano polynomials

Leonard Mada, Maria Anastasia Jivulescu

TL;DR

The paper addresses solving a broad subclass of higher-order polynomials by generalizing Cardano’s cubic method to odd orders using two real parameters $c$ and $d$. It develops an algebraic construction with $p,q$ satisfying $pq=c$ and $p^n+q^n=2d$, yielding roots $x[j]=p\omega^j+q\omega^{-j}$ and a corresponding polynomial $C_{n,c,d}(x)$ with coefficients $B_{m,j}$; it then lifts the construction to an operator framework via the Fujii operator $W=pZ_n+qZ_n^{-1}$ and its Fourier-conjugate Cardano operator $X=F_n^+WF_n$, linking spectral properties to the roots. The approach unifies classical algebra with quantum-information-inspired operator calculus, exposing connections to generalized Chebyshev (via $\Omega_n$) and Ferrari equations, and extends to a Ferrari-operator formulation for quartics. The results illuminate how circulant and Fourier-analytic structures encode polynomial solvability and suggest potential quantum-circuit realizations for spectral polynomial transformations. Overall, the work provides a principled, spectrally grounded framework for a two-parameter family of odd-degree polynomials with practical implications for quantum information and spectral engineering.

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.

An operator algebraic approach for generalized Cardano polynomials

TL;DR

The paper addresses solving a broad subclass of higher-order polynomials by generalizing Cardano’s cubic method to odd orders using two real parameters and . It develops an algebraic construction with satisfying and , yielding roots and a corresponding polynomial with coefficients ; it then lifts the construction to an operator framework via the Fujii operator and its Fourier-conjugate Cardano operator , linking spectral properties to the roots. The approach unifies classical algebra with quantum-information-inspired operator calculus, exposing connections to generalized Chebyshev (via ) and Ferrari equations, and extends to a Ferrari-operator formulation for quartics. The results illuminate how circulant and Fourier-analytic structures encode polynomial solvability and suggest potential quantum-circuit realizations for spectral polynomial transformations. Overall, the work provides a principled, spectrally grounded framework for a two-parameter family of odd-degree polynomials with practical implications for quantum information and spectral engineering.

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.
Paper Structure (14 sections, 4 theorems, 58 equations)

This paper contains 14 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Generalised Cardano polynomials
  3. Algebraic approach to Cardano polynomials.
  4. 3-rd and 5-th order Cardano polynomials
  5. The n-th order Cardano polynomial
  6. Negative discriminant case
  7. Generalized Chebyshev Polynomials
  8. Ferrari polynomial equation.
  9. Cardano operator
  10. Fujii operator
  11. Cardano operator.
  12. Ferrari operator.
  13. Conclusions
  14. Appendix

Key Result

Lemma 2.1

Given $c$ and $d,$ two real parameters( $D:=d^2-c^n\geq 0$) and $n,$ an odd natural number, the system of equations has the set of solutions $\{(p\omega^{j},\,q\omega^{-j})\},\, j\in\{0,\pm 1,\ldots \pm [(n-1)/2]\},$ where and $\omega=e^{\frac{2\pi i}{n}}$ is the n-th order root of unity, ($\omega^n=1$).

Theorems & Definitions (9)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof