Exact linear modeling with polynomial coefficients

Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algebra and the other one works directly in the Weyl algebra, thus requiring the use of tools from non-commutative computer algebra. In behavioral systems theory, the resulting model for the data is known as the most powerful unfalsified model (MPUM) within the class of linear systems with kernel representations over the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients.