ar X iv : 0 90 6 . 46 02 v 1 [ cs . I T ] 2 5 Ju n 20 09 Minimal Gröbner bases and the predictable leading monomial property
نویسندگان
چکیده
In this paper we focus on Gröbner bases over rings for the univariate case. We identify a useful property of minimal Gröbner bases, that we call the " predictable leading monomial (PLM) property ". The property is stronger than " row reducedness " and is crucial in a range of applications. The first part of the paper is tutorial in outlining how the PLM property enables straightforward solutions to classical realization problems of linear systems over fields. In the second part of the paper we use the ideas of [20] on polynomial matrices over the finite ring Zpr (with p a prime integer and r a positive integer) in the more general setting of Gröbner bases and introduce the notion of " Gröbner p-basis " to achieve a predictable leading monomial property over Zpr. This theory finds applications in error control coding over Zpr. Through this approach we are extending the ideas of [20] to a more general context where the user chooses an ordering of polynomial vectors.
منابع مشابه
Minimal Gröbner bases and the predictable leading monomial property
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 deg...
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