We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity 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 ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.
We present two automatic program analyses. The first analysis checks if a given polynomial relation holds among the program variables whenever control reaches a given program point. It fully interprets assignment statements with polynomial expressions on the right-hand side and polynomial disequality guards. Other assignments are treated as non-deterministically assigning any value and guards that are not polynomial disequalities are ignored. The second analysis extends this checking procedure. It computes the set of all polynomial relations of an arbitrary given form that are valid at a given target program point. It is also complete up to the abstraction described above.