Products, polynomials and differential equations in the stream calculus

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 function of the streams 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 resulting unique final coalgebra morphism from polynomials into streams is the (unique) commutative K-algebra homomorphism – and vice versa. This implies that one can algebraically reason 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. Finally, we extend this algorithm to solve a more general problem: finding all valid polynomial equalities that fit in a user specified polynomial template.