Difference between revisions of "ApCoCoA-1:NCo.BInterreduction"

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(New page: <command> <title>NCo.BInterreduction</title> <short_description> Interreduce a LIST of polynomials in a free monoid ring over the binary field. <par/> Note that, given a word ordering, a ...)
 
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{{Version|1}}
 
<command>
 
<command>
 
<title>NCo.BInterreduction</title>
 
<title>NCo.BInterreduction</title>
 
<short_description>
 
<short_description>
 
Interreduce a LIST of polynomials in a free monoid ring over the binary field.  
 
Interreduce a LIST of polynomials in a free monoid ring over the binary field.  
<par/>
 
Note that, given a word ordering, a set <tt>G</tt> of non-zero polynomials is called <em>interreduced</em> if no element of <tt>Supp(g)</tt> is contained in the leading word ideal <tt>LW(G\{g})</tt> for all <tt>g</tt> in <tt>G</tt>.
 
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Note that, given a word ordering, a set <tt>G</tt> of non-zero polynomials is called <em>interreduced</em> if no element of <tt>Supp(g)</tt> is contained in the leading word ideal <tt>BLW(G\{g})</tt> for all <tt>g</tt> in <tt>G</tt>.
 +
<par/>
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
<par/>
 
<par/>
Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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Please set ring environment <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>word ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
 
<itemize>
 
<itemize>
<item>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</tt>. Each polynomial is represented as a LIST of words (or terms) in <tt>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, and the identity is represented as the empty string <quotes></quotes>. Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[<quotes>xy</quotes>, <quotes>y</quotes>, <quotes></quotes>]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
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<item>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</tt>. Each polynomial is represented as a LIST of words (or terms) in <tt>&lt;X&gt;</tt>. Each word is represented as a STRING. For example, <tt>xy^2x</tt> is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial <tt>f=xy-y+1</tt> is represented as F:=["xy", "y", ""]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
 
<item>@return: a LIST of interreduced polynomials.</item>
 
<item>@return: a LIST of interreduced polynomials.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>abc</quotes>);
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NCo.SetX("abc");
NCo.SetOrdering(<quotes>ELIM</quotes>);
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NCo.SetOrdering("ELIM");
Polynomials:=[[<quotes>ba</quotes>,<quotes>c</quotes>],[<quotes>b</quotes>,<quotes></quotes>], [<quotes>c</quotes>]];
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Polynomials:=[["ba","c"],["b",""], ["c"]];
 
NCo.BInterreduction(Polynomials);
 
NCo.BInterreduction(Polynomials);
  
[[<quotes>a</quotes>], [<quotes>b</quotes>, <quotes></quotes>], [<quotes>c</quotes>]]
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[["a"], ["b", ""], ["c"]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.BLW|NCo.BLW</see>
<see>NCo.SetX</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
 +
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<key>NCo.BInterreduction</key>
 
<key>NCo.BInterreduction</key>
 
<key>BInterreduction</key>
 
<key>BInterreduction</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_gbmr</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:37, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.BInterreduction

Interreduce a LIST of polynomials in a free monoid ring over the binary field.

Syntax

NCo.BInterreduction(G:LIST):LIST

Description

Note that, given a word ordering, a set G of non-zero polynomials is called interreduced if no element of Supp(g) is contained in the leading word ideal BLW(G\{g}) for all g in G.

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment alphabet (or set of indeterminates) X and word ordering via the functions NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param G: a LIST of polynomials in the free monoid ring F_{2}<X>. Each polynomial is represented as a LIST of words (or terms) in <X>. Each word is represented as a STRING. For example, xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the polynomial f=xy-y+1 is represented as F:=["xy", "y", ""]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a LIST of interreduced polynomials.

Example

NCo.SetX("abc");
NCo.SetOrdering("ELIM");
Polynomials:=[["ba","c"],["b",""], ["c"]];
NCo.BInterreduction(Polynomials);

[["a"], ["b", ""], ["c"]]
-------------------------------

See also

NCo.BLW

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer