Products, polynomials and differential equations in the stream calculus

We study connections among polynomials, differential equations and streams over a field \(\mathbb {K} \) , 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 \(\mathbb {K} \) -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.