Gröbner bases and behaviors over finite rings

Abstract

For several decades Gröbner bases have proved useful tools for different areas in system theory, particularly multidimensional system theory. These areas range from controller design to minimal realizations of linear systems over fields. In this paper we focus on the univariate case and identify the so-called "predictable leading monomial property" as a property of a minimal Gröbner basis that is crucial in many of these areas. The property is stronger than "row reducedness". We revisit the recently developed theory of [17] in which row reducedness is extended to polynomial matrices over the finite ring Zpr (with p a prime integer and r a positive integer), which find applications in error c..