Gröbner bases and behaviors over finite rings

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 ℤ<inf>p</inf>^r (with p a prime integer and r a positive integer), which find applications in error control coding over ℤ<inf>p</inf>^r. We recast the ideas of [17] in the more general setting of Gröbner bases and derive new results on how to use minimal Gröbner bases to achieve the predictable leading monomial property over ℤ<inf>p</inf>^r. A major advantage of the Gröbner approach is that computational packages are available to compute a minimal Gröbner basis over ℤ<inf>p</inf>^r, such as the SINGULAR computer algebra system. Another advantage of the Gröbner approach is its generality with respect to the choice of ordering of polynomial vectors.