Minimal Gr\"obner bases and the predictable leading monomial property

TL;DR

This paper investigates the properties of minimal Gr"obner bases for modules over rings, introduces the PLM property, and develops methods to obtain bases with this property over finite rings, with applications in coding theory.

Contribution

It introduces the PLM property for minimal Gr"obner bases over rings and provides a method to derive bases with this property over finite rings, enhancing applications in coding.

Findings

PLM property is shared by minimal Gr"obner bases in F[x]^q.

Over rings with zero divisors, minimal Gr"obner bases may lack PLM.

Derived minimal Gr"obner p-bases possess the PLM property and aid in coding applications.

Abstract

We focus on Gr\"obner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the "predictable leading monomial (PLM) property" that is shared by minimal Gr\"{o}bner bases of modules in F[x]^q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70's. Because of the presence of zero divisors, minimal Gr\"{o}bner 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 necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gr\"{o}bner basis, a so-called "minimal Gr\"{o}bner p-basis" that does have a PLM property. We demonstrate that minimal Gr\"obner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z_p^r. Applications are in coding and sequences over Z_p^r.