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TitleMinimal Gröbner bases and the predictable leading monomial property
Author(s) M. Kuijper, Kristina Schindelar
TypeArticle in Journal
AbstractWe 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 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 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 Z p r . Applications are in coding and sequences over Z p r .
KeywordsFinite ring, Polynomial vector modulue, Positional term order, Minimal Gröbner basis, Shortest linear recurrence relation, Parametrization
URL http://www.sciencedirect.com/science/article/pii/S0024379510004386
JournalLinear Algebra and its Applications
Pages104 - 116
Translation No
Refereed No