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

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</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
NC.LC(F:LIST):K
+
NC.LC(F:LIST):RAT
 
</syntax>
 
</syntax>
 
<description>
 
<description>
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Before calling the function, please set ring environment coefficient field <tt>K</tt>, alphabet <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime) (or <ref>NC.UnsetFp</ref>()), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(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.
 
Before calling the function, please set ring environment coefficient field <tt>K</tt>, alphabet <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime) (or <ref>NC.UnsetFp</ref>()), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(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.
 
<itemize>
 
<itemize>
<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 [c, w] where c is in <tt>K</tt> and w is a word in <tt>X*</tt>.  Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. <tt>0</tt> polynomial is represented as an empty LIST []. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</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 c is in <tt>K</tt> and w is a word in <tt>X*</tt>.  Unit in <tt>X*</tt> is empty word represented as an empty string <quotes></quotes>. <tt>0</tt> polynomial is represented as an empty list. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</item>
<item>@return: an element of K which is leading term of <tt>F</tt> with respect to current ordering. If <tt>F=0</tt>, then return <tt>0</tt>. </item>
+
<item>@return: an element of the current field which is the leading term of <tt>F</tt> with respect to the current ordering. If <tt>F=0</tt>, then <tt>0</tt> will be returned. </item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>

Revision as of 14:13, 14 October 2010

NC.LC

Leading coefficient of a polynomial over a free associative K-algebra.

Syntax

NC.LC(F:LIST):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.

Before calling the function, please set ring environment coefficient field K, alphabet X and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is Q. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param F: a polynomial in K<X>. Each polynomial in K<X> is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in K and w is a word in X*. Unit in X* is empty word represented as an empty string "". 0 polynomial is represented as an empty list. For example, polynomial F:=xy-y+1 in K<x,y> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

  • @return: an element of the current field which is the leading term of F with respect to the current ordering. If F=0, then 0 will be returned.

Example

NC.SetX(<quotes>abc</quotes>);
F:=[[1,<quotes>ab</quotes>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<quotes>bab</quotes>]];
NC.SetOrdering(<quotes>ELIM</quotes>);
NC.LC(F);
2
-------------------------------
NC.SetOrdering(<quotes>LLEX</quotes>);
NC.LC(F);
4
-------------------------------
NC.LC([]); -- 0 polynomial
0
-------------------------------

See also

NC.Add

NC.BP

NC.Deg

NC.FindPolynomials

NC.GB

NC.Intersection

NC.IsGB

NC.KernelOfHomomorphism

NC.LT

NC.LTIdeal

NC.MRAdd

NC.MRBP

NC.MRIntersection

NC.MRKernelOfHomomorphism

NC.MRMinimalPolynomials

NC.MRMultiply

NC.MRReducedBP

NC.MRSubtract

NC.MinimalPolynomial

NC.Multiply

NC.NR

NC.ReducedBP

NC.ReducedGB

NC.SetFp

NC.SetOrdering

NC.SetRelations

NC.SetRules

NC.SetX

NC.Subtract

NC.UnsetFp

NC.UnsetOrdering

NC.UnsetRelations

NC.UnsetRules

NC.UnsetX

Introduction to CoCoAServer