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Mathematics in Computer Science

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17.12.2019

Bar Code: A Visual Representation for Finite Sets of Terms and Its Applications

verfasst von: Michela Ceria

Erschienen in: Mathematics in Computer Science | Ausgabe 2/2020

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Abstract

The Bar Code is a bidimensional diagram representing a finite set of terms in any number of variables. In particular, one can represent the (lexicographical) Groebner escalier of a zerodimensional monomial ideal and use this representation to desume many of its properties. The aim of this paper is to give a general description of the Bar Code and it construction, giving then an overview of all the applications studied so far.

Vorheriger Artikel An Algorithm for Computing Grothendieck Local Residues II: General Case

Nächster Artikel A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

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Actually, it can be easily observed that \({\textsf {T}}(x_1^2-x_1)=x_1^2\), \({\textsf {T}}(x_1x_2)=x_1x_2\), \({\textsf {T}}(x_2^2-2x_2)=x_2^2\) trivially holds for each term order.

Actually, in this context, “high-dimensional” means “of dimension greater than or equal to” 4.

Clearly if a term \(\pi ^i(\tau _{\overline{j}})\) is not repeated in \({\overline{M}}^{[i]}\), the sublist containing it will be only \([\pi ^i(\tau _{\overline{j}})]\), i.e. \(h=0\).

Notice that these assignments are those given by \(\mathfrak {BC}1\) and \(\mathfrak {BC}2\).

The Lex game gives the same escalier, but it is not so focused on the bijection between \({\mathbf {X}}\) and \({\textsf {N}}(I({\mathbf {X}}))\).

For an efficient implementation, see [11, 12].

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Titel: Bar Code: A Visual Representation for Finite Sets of Terms and Its Applications
verfasst von: Michela Ceria
Publikationsdatum: 17.12.2019
Verlag: Springer International Publishing
Erschienen in: Mathematics in Computer Science / Ausgabe 2/2020
Print ISSN: 1661-8270
Elektronische ISSN: 1661-8289
DOI: https://doi.org/10.1007/s11786-019-00425-4

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