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 control coding over Zpr . 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 Zpr . A major advantage of the Gröbner approach is that computational packages are available to compute a minimal Gröbner basis over Zpr , 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.

On Minimality of Convolutional Ring Encoders

“…In recent work [14] it is shown that computational packages for computing minimal Gröbner bases can be used to construct a minimal p-encoder.…”

Minimal Gröbner bases and the predictable leading monomial property

“…Finally, a third advantage of the Gröbner approach is that computational packages are available to compute minimal Gröbner bases, such as the Singular computer algebra system [10]. A preliminary version of this paper is [16].…”