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

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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>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 the 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 the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
 
<itemize>
 
<itemize>
<item>@param <em>F:</em> a polynomial 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>F:</em> a polynomial 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>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</tt>.</item>
 
<item>@param <em>G:</em> a LIST of polynomials in the free monoid ring <tt>F_{2}&lt;X&gt;</tt>.</item>
 
<item>@return: a LIST which represents the normal remainder of F with respect to G.</item>
 
<item>@return: a LIST which represents the normal remainder of F with respect to G.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NCo.SetX(<quotes>xyXY</quotes>);
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NCo.SetX("xyXY");
NCo.SetOrdering(<quotes>LLEX</quotes>);
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NCo.SetOrdering("LLEX");
F1:=[<quotes>xX</quotes>,<quotes></quotes>];
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F1:=["xX",""];
F2:=[<quotes>Xx</quotes>,<quotes></quotes>];
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F2:=["Xx",""];
F3:=[<quotes>yY</quotes>,<quotes></quotes>];
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F3:=["yY",""];
F4:=[<quotes>Yy</quotes>,<quotes></quotes>];
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F4:=["Yy",""];
 
G:=[F1,F2,F3,F4];
 
G:=[F1,F2,F3,F4];
F:=[<quotes>xyYxxXYY</quotes>];
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F:=["xyYxxXYY"];
 
NCo.BNR(F,G);
 
NCo.BNR(F,G);
[<quotes>xxYY</quotes>]
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["xxYY"]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>

Latest revision as of 13:38, 29 October 2020

This article is about a function from ApCoCoA-1.

NCo.BNR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring over the binary field F_{2}={0,1}.

Syntax

NCo.BNR(F:LIST, G:LIST):INT

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 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 the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param F: a polynomial 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 [].

  • @param G: a LIST of polynomials in the free monoid ring F_{2}<X>.

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

Example

NCo.SetX("xyXY");
NCo.SetOrdering("LLEX");
F1:=["xX",""];
F2:=["Xx",""];
F3:=["yY",""];
F4:=["Yy",""];
G:=[F1,F2,F3,F4];
F:=["xyYxxXYY"];
NCo.BNR(F,G);
["xxYY"]
-------------------------------

See also

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer