  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.
