Explicit geometric integration of polynomial vector fields

We present a unified framework in which to study splitting methods for polynomial vector fields in R. The vector field is to be represented as a sum of shears, each of which can be integrated exactly, and each of which is a function of k < n variables. Each shear must also inherit the structure of the original vector field: we consider Hamiltonian, Poisson, and volumepreserving cases. Each case then leads to the problem of finding an optimal distribution of points on an appropriate homogeneous space, generally the Grassmannians of k-planes or (in the Hamiltonian case) isotropic k-planes in R. These optimization problems have the same structure as those of constructing optimal experimental designs in statistics.