标题： 代数框架用于洛朗级数上的离散动力系统 显示英文标题

标题： Algebraic Framework for Discrete Dynamical Systems over Laurent Series

摘要： 我们推广了离散代数动力系统的框架\cite{Andriamifidisoa2014}，使其适用于\(\Z^r\)上的Laurent多项式和级数，从而能够对双向离散系统进行建模。 通过重新定义空间\(\Dprime\)和\(\Aprime\)，引入一个双线性映射（在第3节中定义为标量积），并扩展移位算子，我们保留了\cite{Andriamifidisoa2014}的对偶性和伴随性质。 这些性质通过示例和一个关于双向序列变换的数据处理案例研究进行了严格证明和说明。 与 Oberst\cite{Ob90}相比，我们的代数方法强调Laurent级数的结构，为多维系统提供了一个简化的框架。 这项工作解决了\cite{Andriamifidisoa2014}提出的一个开放问题，并在多维数据处理中有所应用，例如图像滤波和控制理论。 显示英文摘要

摘要： We generalize the framework of discrete algebraic dynamical systems \cite{Andriamifidisoa2014} to Laurent polynomials and series over \(\Z^r\), enabling the modeling of bidirectional discrete systems. By redefining the spaces \(\Dprime\) and \(\Aprime\), introducing a bilinear mapping (defined as the scalar product in Section 3), and extending the shift operator, we preserve the duality and adjoint properties of \cite{Andriamifidisoa2014}. These properties are rigorously proved and illustrated through examples and a data processing case study on bidirectional sequence transformations. In contrast to Oberst \cite{Ob90}, our algebraic approach emphasizes the structure of Laurent series, providing a streamlined framework for multidimensional systems. This work addresses an open question from \cite{Andriamifidisoa2014} and has applications in multidimensional data processing, such as image filtering and control theory.