Affiliation:
1. Università degli Studi di Firenze, Italy
2. Università di Roma “La Sapienza”, Italy
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 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.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Mathematics,Logic,General Computer Science,Theoretical Computer Science
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