Algebra and coalgebra of stream products
Michele Boreale, Daniele Gorla
TL;DR
The paper develops a unified algebraic-coalgebraic framework linking polynomials, differential equations, and streams through $(F,G)$-products on streams. By equipping polynomials with a dependent transition $\delta_\pi$ that mirrors a given stream product, it shows that the unique final coalgebra morphism $\mu_\pi$ to streams is simultaneously a $\mathbb{K}$-algebra homomorphism, enabling algebraic reasoning about streams. The authors provide an algebraic-geometric algorithm to decide polynomial-stream equivalence and demonstrate how this framework yields generating-function analyses and ODE solutions for concrete products such as convolution, shuffle, and Hadamard. They also connect algebraicity and analytic properties of streams to classical tools like Gröbner bases, Lie derivatives, and Laplace-transform-like relations, offering a pathway to closed-form generating functions and non-linear ODE solutions. The work emphasizes full abstraction and algorithmic decision procedures, with potential extensions to new products and deeper links to bialgebra theory.
Abstract
We study connections among polynomials, differential equations and streams over a field K, in terms of algebra and coalgebra. We first introduce the class of (F,G)-products on streams, those where the stream derivative of a product can be expressed as a polynomial of the streams themselves and their derivatives. Our first result is that, for every (F,G)-product, there is a canonical way to construct a transition function on polynomials such that the induced unique final coalgebra morphism from polynomials into streams is the (unique) K-algebra homomorphism -- and vice versa. This implies one can reason algebraically on streams, via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G)-product. As an example of reasoning on streams, we focus on specific products (convolution, shuffle, Hadamard) and show how to obtain closed forms of algebraic generating functions of combinatorial sequences, as well as solutions of nonlinear ordinary differential equations.