Divided powers and composition in integro-differential algebras

Abstract

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.