Minimal Gröbner bases and the predictable leading monomial property |
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Authors: | M Kuijper K Schindelar |
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Institution: | a Department of Electrical and Electronic Engineering, University of Melbourne, Vic 3010, Australia b Lehrstuhl D für Mathematik, RWTH Aachen University Templergraben 64, 52062 Aachen, Germany |
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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 Fx]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. |
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Keywords: | Finite ring Polynomial vector modulue Positional term order Minimal Grö bner basis Shortest linear recurrence relation Parametrization |
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