ApCoCoA-1:NC.LC: Difference between revisions

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{{Version|1}}
<command>
<command>
<title>NC.LC</title>
<title>NC.LC</title>
<short_description>
<short_description>
Leading coefficient of a polynomial over a free monoid ring.
Leading coefficient of a non-zero polynomial in a non-commutative polynomial ring.
</short_description>
</short_description>
<syntax>
<syntax>
Line 10: 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 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 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 an 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 INT or RAT which represents the leading coefficient of F w.r.t. the current ordering. If <tt>F=0</tt>, the function returns <tt>0</tt>.</item>
<item>@return: an INT or a RAT, whhich is the leading coefficient 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>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<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.SetOrdering(<quotes>ELIM</quotes>);
NC.SetOrdering("LLEX");
NC.LC(F);
NC.LC(F);
2


[1]
-------------------------------
-------------------------------
NC.SetOrdering(<quotes>LLEX</quotes>);
NC.SetOrdering("LRLEX");
NC.LC(F);
NC.LC(F);
4


[4]
-------------------------------
-------------------------------
NC.LC([]); -- 0 polynomial
NC.SetOrdering("ELIM");
0
NC.LC(F);
 
[1]
-------------------------------
NC.SetOrdering("DEGRLEX");
NC.LC(F);


[1]
-------------------------------
-------------------------------
</example>
</example>
</description>
</description>
<seealso>
<seealso>
<see>NC.Add</see>
<see>ApCoCoA-1:Use|Use</see>
<see>NC.Deg</see>
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>NC.FindPolynomials</see>
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
<see>NC.GB</see>
<see>NC.HF</see>
<see>NC.Interreduction</see>
<see>NC.Intersection</see>
<see>NC.IsGB</see>
<see>NC.IsHomog</see>
<see>NC.KernelOfHomomorphism</see>
<see>NC.LC</see>
<see>NC.LT</see>
<see>NC.LTIdeal</see>
<see>NC.MB</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.LC</key>
<key>ncpoly.LC</key>
<key>NC.LC</key>
<key>NC.LC</key>
<key>LC</key>
<key>LC</key>
<wiki-category>Package_gbmr</wiki-category>
<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
</command>
</command>

Latest revision as of 13:34, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.LC

Leading coefficient of a non-zero polynomial in a non-commutative polynomial ring.

Syntax

NC.LC(F:LIST):INT or RAT

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: an INT or a RAT, whhich is the leading coefficient 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.LC(F);

[1]
-------------------------------
NC.SetOrdering("LRLEX");
NC.LC(F);

[4]
-------------------------------
NC.SetOrdering("ELIM");
NC.LC(F);

[1]
-------------------------------
NC.SetOrdering("DEGRLEX");
NC.LC(F);

[1]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer




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