Algebra, Coalgebra, and Minimization in Polynomial Differential Equations

We consider reasoning and minimization in systems of polynomial ordinary differential equations (odes). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow polynomials with a transition system structure based on the concept of Lie derivative, thus inducing a notion ofL-bisimulation. Two states (variables) are provenL-bisimilar if and only if they correspond to the same solution in the odes system. We then characterize Lbisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of odes, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations.