Laurent Sequences, Extended Rota Algebras and Categorical Discretization of Dynamical Systems

Abstract

We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that enables a unified treatment of continuous and discrete dynamical systems. Our theoretical framework is grounded on a generalization of G. C. Rota's finite operator calculus, which enables us to extend the theory of basic sequence of polynomials to the setting of Laurent polynomials. Accordingly, we introduce the notion of an \textit{extended Rota algebra}, defined as a Galois differential algebra in which all difference operators act as derivations on the space of Laurent power series with respect to a suitably defined functional product. The core of our theory relies on the existence of covariant functors between the newly proposed Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. In this setting, under certain regularity assumptions, a differential equation and its discrete analogues are naturally interpreted as objects of the same category. This perspective enables the construction of a vast class of integrable maps that share with their continuous analogues a wide set of exact solutions, \textit{regular} or \textit{singular} and, in the linear case, the Picard-Vessiot group.