一个算子代数方法用于广义卡丹多项式
An operator algebraic approach for generalized Cardano polynomials
AI总结：
本文通过算子代数方法研究广义卡丹多项式，揭示其与量子信息理论的关联，并探讨其在多项式求解中的应用。
摘要：
我们开发了一个用于广义卡丹多项式的算子代数框架，并展示了其结构自然导致一种与量子信息理论工具和概念兼容的卡丹方法的算子表述。广义卡丹多项式作为经典卡丹公式解三次方程理论的推广，以及通过圆算子的谱性质构建，将卡丹类型恒等式嵌入其谱理论中。该构造澄清了两参数奇数阶多项式族的代数结构和可解性，经典方法和量子信息理论中熟悉的傅里叶变换和算子代数上的谱算子理论。作为应用，我们展示了与切比雪夫多项式以及四次方程费拉里解法的联系。
英文摘要：
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 a circular operator W, expressed in terms of n dimensional clock operator. This representation embeds Cardano type identities into the spectral theory of circulant operators. 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. As application, spectral calculus on operator algebras is used to construct Cardano generated POVMs.
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