Geometric Properties of Over-Determined Systems of Linear Partial Difference Equations

We relate linear constant coefficient systems of partial difference equations (a discretization of a system of linear partial differential equations) satisfying some collection of scalar polynomial equations to systems defined over the coordinate ring of an algebraic variety. This motivates the extension of behavioral systems theory (a generalization of classical systems theory where inputs and outputs are lumped together) to the setting where the ring of operators is an affine domain and the signal space is restricted to signals which satisfy the same scalar polynomial equations. By recognizing the role of the kernel representation's Gröbner basis in the Cauchy problem, we extend notions of controllability from the classical behavioral setting to accommodate this generalization. We then address the question as to when an autonomous behavior admits a Livšic-system state-space representation, where the state update equations are overdetermined leading to the requirement that the input and output signals satisfy their own compatibility difference equations. This leads to a frequency domain setting involving input and output holomorphic vector bundles and a transfer function given by a meromorphic bundle map. An analogue of the Hankel realization theorem developed by J. Ball and V. Vinnikov then leads to a Livšic-system state-space representation for an autonomous behavior satisfying some natural additional conditions.