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

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<command>
 
<command>
 
<title>NCo.Interreduction</title>
 
<title>NCo.Interreduction</title>
 
<short_description>
 
<short_description>
 
Interreduce a LIST of polynomials in a free monoid ring.  
 
Interreduce a LIST of polynomials in a free monoid ring.  
<par/>
 
Note that, given a word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called <em>interreduced</em> with respect to <tt>Ordering</tt> 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 <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called <em>interreduced</em> with respect to <tt>Ordering</tt> 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>.
 +
<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>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
 
<itemize>
 
<itemize>
<item>@param <em>G</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<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 <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
 
<item>@return: a LIST of interreduced polynomials with respect to the current word ordering.</item>
 
<item>@return: a LIST of interreduced polynomials with respect to the current word ordering.</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");
G:=[[[1,<quotes>ba</quotes>]], [[1,<quotes>b</quotes>],[1,<quotes></quotes>]], [[1,<quotes>c</quotes>]]];
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G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]];
 
NCo.Interreduction(G);
 
NCo.Interreduction(G);
  
[[[1, <quotes>a</quotes>]], [[1, <quotes>b</quotes>], [1, <quotes></quotes>]], [[1, <quotes>c</quotes>]]]
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[[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NCo.LW</see>
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<see>ApCoCoA-1:NCo.LW|NCo.LW</see>
<see>NCo.SetFp</see>
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<see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see>
<see>NCo.SetOrdering</see>
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<see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see>
<see>NCo.SetX</see>
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<see>ApCoCoA-1:NCo.SetX|NCo.SetX</see>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<key>NCo.Interreduction</key>
 
<key>NCo.Interreduction</key>
 
<key>Interreduction</key>
 
<key>Interreduction</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 13:39, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.Interreduction

Interreduce a LIST of polynomials in a free monoid ring.

Syntax

NCo.Interreduction(G:LIST):LIST

Description

Note that, given a word ordering Ordering, a set of non-zero polynomials G is called interreduced with respect to Ordering if no element of Supp(g) is contained in the leading word ideal LW(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 coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param G: a LIST of polynomials in K<X>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a LIST of interreduced polynomials with respect to the current word ordering.

Example

NCo.SetX("abc");
NCo.SetOrdering("ELIM");
G:=[[[1,"ba"]], [[1,"b"],[1,""]], [[1,"c"]]];
NCo.Interreduction(G);

[[[1, "a"]], [[1, "b"], [1, ""]], [[1, "c"]]]
-------------------------------

See also

NCo.LW

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer