Minimal Gröbner bases and the predictable leading monomial property

Abstract

We focus on Gröbner 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 Gröbner 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 Gröbner bases over a finite ring of the type Zpr (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 Gröbner basis, a so-called "minimal Gröbner p-basis" that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Zpr . Applications are in coding and sequences over Zpr . * M. Kuijper is with the

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“…As a result, it also holds over the ring of linearized polynomials. It was labeled Predictable Leading Monomial (PLM) property in [13] to emphasize its closeness to Forney's Predictable Degree property [8]. It captures the exact property that is needed in subsequent proofs.…”

“…Note that in [13] minimal bases were addressed as minimal Gröbner bases. It can be shown that in our current setting these are the same.…”

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“…Therefore, a necessary condition for the existence of a feasible solution satisfying both the constraints (18) and (21) is that L intersects C at two different points (M 1 , ρ 1 ) and (M 2 , ρ 2 ) on the real plane. Now solving (25) and (26) for M we get…”

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