Algebraic Framework for Discrete Dynamical Systems over Laurent Series

2025/07/15 by Aandriamifidisoa, Ramamonjy, Saindou, Loukman Ben
#13C60 #18A40 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2508.04708

Abstract

We generalize the framework of discrete algebraic dynamical systems \citeAndriamifidisoa2014 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 \citeAndriamifidisoa2014. These properties are rigorously proved and illustrated through examples and a data processing case study on bidirectional sequence transformations. In contrast to Oberst \citeOb90, our algebraic approach emphasizes the structure of Laurent series, providing a streamlined framework for multidimensional systems. This work addresses an open question from \citeAndriamifidisoa2014 and has applications in multidimensional data processing, such as image filtering and control theory.

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