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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.NR</title>
 
<title>NC.NR</title>
 
<short_description>
 
<short_description>
Normal remainder polynomial with respect to a list of polynomials over a free associative <tt>K</tt>-algebra.
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Normal remainder of a polynomial with respect to a LIST of polynomials in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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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 coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.
 
<itemize>
 
<itemize>
<item></item>
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<item>@param <em>F</em>: a non-commutative polynomial. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item>
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<item>@param <em>G</em>: a LIST of non-zero non-commutative polynomials.</item>
<item>@param <em>G</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item>
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<item>@return: a LIST, which is the normal remainder of F with respect to G.</item>
<item>@return: a LIST which represents a normal remainder of F with respect to G.</item>
 
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NC.SetX(<quotes>abc</quotes>);
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USE QQ[x[1..2],y[1..2]];
NC.RingEnv();
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NC.SetOrdering("LLEX");
Coefficient ring : Q
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F:= [[x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3],[5]]; -- x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5
Alphabet : abc
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G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
Ordering : LLEX
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G2:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2]
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NC.NR(F,[G1,G2]);
  
-------------------------------
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[[-9y[2], x[1]^2, x[2]^3], [-x[1], y[2], x[2]^2], [5]]
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];
 
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]];
 
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
 
G:=[F1,F2];
 
NC.NR(F,G);
 
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]
 
-------------------------------
 
NC.SetOrdering(<quotes>ELIM</quotes>);
 
NC.NR(F,G);
 
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NC.Add</see>
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<see>ApCoCoA-1:Use|Use</see>
<see>NC.Deg</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>NC.FindPolynomials</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
<see>NC.GB</see>
 
<see>NC.HF</see>
 
<see>NC.Intersection</see>
 
<see>NC.IsGB</see>
 
<see>NC.KernelOfHomomorphism</see>
 
<see>NC.LC</see>
 
<see>NC.LT</see>
 
<see>NC.LTIdeal</see>
 
<see>NC.MinimalPolynomial</see>
 
<see>NC.Multiply</see>
 
<see>NC.NR</see>
 
<see>NC.ReducedGB</see>
 
<see>NC.SetFp</see>
 
<see>NC.SetOrdering</see>
 
<see>NC.SetRelations</see>
 
<see>NC.SetRules</see>
 
<see>NC.SetX</see>
 
<see>NC.Subtract</see>
 
<see>NC.UnsetFp</see>
 
<see>NC.UnsetOrdering</see>
 
<see>NC.UnsetRelations</see>
 
<see>NC.UnsetRules</see>
 
<see>NC.UnsetX</see>
 
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<type>non_commutative</type>
 
<type>non_commutative</type>
 
</types>
 
</types>
<key>gbmr.NR</key>
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<key>ncpoly.NR</key>
 
<key>NC.NR</key>
 
<key>NC.NR</key>
 
<key>NR</key>
 
<key>NR</key>
<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:35, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.NR

Normal remainder of a polynomial with respect to a LIST of polynomials in a non-commutative polynomial ring.

Syntax

NC.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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

  • @param F: a non-commutative polynomial. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST [].

  • @param G: a LIST of non-zero non-commutative polynomials.

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

Example

USE QQ[x[1..2],y[1..2]];
NC.SetOrdering("LLEX");
F:= [[x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3],[5]]; -- x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5
G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
G2:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2]
NC.NR(F,[G1,G2]);

[[-9y[2], x[1]^2, x[2]^3], [-x[1], y[2], x[2]^2], [5]]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer