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

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<command>
 
<command>
 
<title>NCo.NR</title>
 
<title>NCo.NR</title>
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<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></item>
 
<item></item>
<item>@param <em>F</em>: a polynomial 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>F</em>: a polynomial 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>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt>.</item>
 
<item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt>.</item>
 
<item>@return: a LIST which represents the normal remainder of <tt>F</tt> with respect to <tt>G</tt>.</item>
 
<item>@return: a LIST which represents the normal remainder of <tt>F</tt> with respect to <tt>G</tt>.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>abc</quotes>);
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NCo.SetX("abc");
 
NCo.RingEnv();
 
NCo.RingEnv();
 
Coefficient ring : Q
 
Coefficient ring : Q
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Ordering : LLEX
 
Ordering : LLEX
 
-------------------------------
 
-------------------------------
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];
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F:=[[1,"ab"],[1,"aca"],[1,"bb"],[1,"bab"],[1,""]];
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]];  
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F1 := [[1,"a"],[1,"c"]];  
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
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F2 := [[1,"b"],[1,"ba"]];
 
G:=[F1,F2];
 
G:=[F1,F2];
 
NCo.NR(F,G);
 
NCo.NR(F,G);
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]
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[[-1,"bcb"], [1,"ccc"], [1,"bb"], [-1,"cb"], [1,""]]
 
-------------------------------
 
-------------------------------
NCo.SetOrdering(<quotes>ELIM</quotes>);
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NCo.SetOrdering("ELIM");
 
NCo.NR(F,G);
 
NCo.NR(F,G);
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]
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[[-1,"bcb"], [1,"bb"], [-1,"cb"], [1,"ccc"], [1,""]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<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.NR</key>
 
<key>NCo.NR</key>
 
<key>NR</key>
 
<key>NR</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 13:43, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.NR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring.

Syntax

NCo.NR(F:LIST, G:LIST):LIST

Description

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 F: a polynomial 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 [].

  • @param G: a LIST of non-zero polynomials in K<X>.

  • @return: a LIST which represents the normal remainder of F with respect to G.

Example

NCo.SetX("abc");
NCo.RingEnv();
Coefficient ring : Q
Alphabet : abc
Ordering : LLEX
-------------------------------
F:=[[1,"ab"],[1,"aca"],[1,"bb"],[1,"bab"],[1,""]];
F1 := [[1,"a"],[1,"c"]]; 
F2 := [[1,"b"],[1,"ba"]];
G:=[F1,F2];
NCo.NR(F,G);
[[-1,"bcb"], [1,"ccc"], [1,"bb"], [-1,"cb"], [1,""]]
-------------------------------
NCo.SetOrdering("ELIM");
NCo.NR(F,G);
[[-1,"bcb"], [1,"bb"], [-1,"cb"], [1,"ccc"], [1,""]]
-------------------------------

See also

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer