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

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(New page: <command> <title>NC.NR</title> <short_description> Normal remainder polynomial with respect to a list of polynomials over a free associative K-algebra. </short_description> <syntax> 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 K-algebra.
+
Normal remainder of a polynomial with respect to a LIST of polynomials in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
−
NC.NR(F:LIST, Polynomials:LIST):LIST
+
NC.NR(F:LIST, G:LIST):LIST
 
</syntax>
 
</syntax>
 
<description>
 
<description>
 
<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/>
 +
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>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item>
+
<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 in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>.  Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item>
+
<item>@param <em>G</em>: a LIST of non-zero non-commutative polynomials.</item>
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<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item>
+
<item>@return: a LIST, which is the normal remainder of F with respect to G.</item>
−
<item>@return: a STRING which represents normal remainder of <tt>F</tt> with respect to <tt>Polynomials</tt>.</item>
 
 
</itemize>
 
</itemize>
 
<example>
 
<example>
−
NC.UnsetFp();
+
USE QQ[x[1..2],y[1..2]];
−
NC.RingEnv();
+
NC.SetOrdering("LLEX");
−
Coefficient ring : Q (float type in C++)
+
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
+
G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
−
Ordering : ELIM
+
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]]
−
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aa</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>]];
 
−
NC.LT(F); -- ELIM ordering
 
−
aa
 
−
-------------------------------
 
−
NC.SetOrdering(<quotes>LLEX</quotes>);
 
−
NC.LT(F); -- LLEX ordering
 
−
bab
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
−
<see>NC.Add</see>
+
<see>ApCoCoA-1:Use|Use</see>
−
<see>NC.LT</see>
+
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
−
<see>NC.Multiply</see>
+
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
−
<see>NC.Subtract</see>
 
−
<see>Gbmr.MRSubtract</see>
 
−
<see>Gbmr.MRMultiply</see>
 
−
<see>Gbmr.MRBP</see>
 
−
<see>Gbmr.MRIntersection</see>
 
−
<see>Gbmr.MRKernelOfHomomorphism</see>
 
−
<see>Gbmr.MRMinimalPolynomials</see>
 
−
<see>Introduction to CoCoAServer</see>
 
 
</seealso>
 
</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
−
<type>groebner</type>
+
<type>polynomial</type>
 +
<type>non_commutative</type>
 
</types>
 
</types>
 +
<key>ncpoly.NR</key>
 
<key>NC.NR</key>
 
<key>NC.NR</key>
 
<key>NR</key>
 
<key>NR</key>
−
<wiki-category>Package_gbmr</wiki-category>
+
<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




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