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The “predictable degree property”, a terminology introduced by Forney in 1970, is a property of polynomial matrices over a field F that has proven itself to be fundamentally useful for a range of applications. In this paper we strengthen this property into the “predictable leading monomial” property, and show that this PLM property is shared by minimal Grobner bases for any positional term order (here: TOP and POT) in F[x]q. The property is useful particularly for minimal interpolationtype problems. Because of the presence of zero divisors, minimal Grobner bases over a finite ring of the type ℤ p r (where p is a prime integer and r is an integer > 1) do not have the PLM property. We show how to construct, from an ordered minimal Grobner basis, a so-called minimal Grobner p-basis that does have a PLM property. The parametrization of all shortest linear recurrence relations of a finite sequence over ℤ p r is a type of problem for which this is useful and we include an illustrative example.

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