A unified pole-zero module for linear transfer functions

A new module-theoretic tool is presented for the study of linear time-invariant input/output maps (‘transfer functions’). For suitable subrings of the field k(s) of rational functions, an O-module called the structure module can be attached to each transfer function. This structure module naturally contains the classical (Kalman) pole module as a submodule which gives the (Wyman-Sain) multivariable zero module as the corresponding factor module. A posteriori the resulting exact sequence splits, and the structure module is isomorphic to the direct sum of the pole module and the zero module. Calculations over local rings echo valuation-theoretic results obtained by Kung and Kailath.