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

From ApCoCoAWiki
m (replaced <quotes> tag by real quotes)
 
(8 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
{{Version|1}}
 
<command>
 
<command>
 
<title>NC.LW</title>
 
<title>NC.LW</title>
 
<short_description>
 
<short_description>
Leading word of a non-zero polynomial in a non-commutative polynomial ring.
+
The leading word (or term) of a non-zero polynomial in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
NC.LW(F:LIST):STRING
+
NC.LW(F:LIST):LIST
 +
NC.LT(F: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/>
 
<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>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. The default coefficient field is <tt>Q</tt>. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant 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 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 polynomial is represented as a LIST of monomials, which are pairs 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>
+
<item>@param <em>F</em>: a non-zero 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>@return: a STRING which represents the leading term of F w.r.t. the current ordering. If <tt>F=0</tt>, the function returns nothing.</item>
+
<item>@return: a LIST, which is the leading word of F with respect to the current word ordering.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NC.SetX(<quotes>abc</quotes>);
+
USE QQ[x[1..2]];
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aa</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>]];
+
F:= [[x[1]^2], [2x[1],x[2]], [3x[2],x[1]],[4x[2]^2]]; -- x[1]^2+2x[1]x[2]+3x[2]x[1]+4x[2]^2
NC.LW(F); -- LLEX ordering (default ordering)
+
NC.SetOrdering("LLEX");
bab
+
NC.LW(F);
 +
 
 +
[x[1]^2]
 
-------------------------------
 
-------------------------------
NC.SetOrdering(<quotes>ELIM</quotes>);
+
-- Done.
NC.LW(F);  
+
-------------------------------
aa
+
NC.SetOrdering("LRLEX");
 +
NC.LW(F);
 +
 
 +
[x[2]^2]
 +
-------------------------------
 +
-- Done.
 +
-------------------------------
 +
NC.SetOrdering("ELIM");
 +
NC.LW(F);
 +
 
 +
[x[1]^2]
 +
-------------------------------
 +
-- Done.
 +
-------------------------------
 +
NC.SetOrdering("DEGRLEX");
 +
NC.LT(F);
 +
 
 +
[x[1]^2]
 
-------------------------------
 
-------------------------------
NC.LW([]);
 
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NC.SetOrdering</see>
+
<see>ApCoCoA-1:Use|Use</see>
<see>Introduction to CoCoAServer</see>
+
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
 +
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
Line 40: Line 61:
 
<key>NC.LW</key>
 
<key>NC.LW</key>
 
<key>LW</key>
 
<key>LW</key>
<wiki-category>Package_ncpoly</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.LW

The leading word (or term) of a non-zero polynomial in a non-commutative polynomial ring.

Syntax

NC.LW(F:LIST):LIST
NC.LT(F: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-zero 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 [].

  • @return: a LIST, which is the leading word of F with respect to the current word ordering.

Example

USE QQ[x[1..2]];
F:= [[x[1]^2], [2x[1],x[2]], [3x[2],x[1]],[4x[2]^2]]; -- x[1]^2+2x[1]x[2]+3x[2]x[1]+4x[2]^2
NC.SetOrdering("LLEX");
NC.LW(F);

[x[1]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering("LRLEX");
NC.LW(F);

[x[2]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering("ELIM");
NC.LW(F);

[x[1]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering("DEGRLEX");
NC.LT(F);

[x[1]^2]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer