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

From ApCoCoAWiki
m (insert version info)
m (replaced <quotes> tag by real quotes)
 
Line 11: Line 11:
 
<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 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 (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions.
+
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>@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 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>
Line 19: Line 19:
 
<example>
 
<example>
 
USE QQ[x[1..2],y[1..2]];
 
USE QQ[x[1..2],y[1..2]];
NC.SetOrdering(<quotes>LLEX</quotes>);
+
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
 
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
 
G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2

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