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

From ApCoCoAWiki
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<title>NC.LC</title>
 
<title>NC.LC</title>
 
<short_description>
 
<short_description>
Leading coefficient of a polynomial over a free associative <tt>K</tt>-algebra.
+
Leading coefficient of a polynomial over a free monoid ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<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 coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
+
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. 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 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 term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item>
+
<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>@return: a INT or RAT which represents a leading coefficient of F. </item>
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<item>@return: a INT or RAT which represents the leading coefficient of F. If <tt>F=0</tt>, the function returns <tt>0</tt>.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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NC.LC(F);
 
NC.LC(F);
 
2
 
2
 +
 
-------------------------------
 
-------------------------------
 
NC.SetOrdering(<quotes>LLEX</quotes>);
 
NC.SetOrdering(<quotes>LLEX</quotes>);
 
NC.LC(F);
 
NC.LC(F);
 
4
 
4
 +
 
-------------------------------
 
-------------------------------
 
NC.LC([]); -- 0 polynomial
 
NC.LC([]); -- 0 polynomial
 
0
 
0
 +
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
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<see>NC.GB</see>
 
<see>NC.GB</see>
 
<see>NC.HF</see>
 
<see>NC.HF</see>
 +
<see>NC.Interreduction</see>
 
<see>NC.Intersection</see>
 
<see>NC.Intersection</see>
 
<see>NC.IsGB</see>
 
<see>NC.IsGB</see>

Revision as of 15:00, 7 June 2012

NC.LC

Leading coefficient of a polynomial over a free monoid 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 ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. 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 is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial 0 is represented as an empty LIST [].

  • @return: a INT or RAT which represents the leading coefficient of F. If F=0, the function returns 0.

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.Deg

NC.FindPolynomials

NC.GB

NC.HF

NC.Interreduction

NC.Intersection

NC.IsGB

NC.KernelOfHomomorphism

NC.LC

NC.LT

NC.LTIdeal

NC.MinimalPolynomial

NC.Multiply

NC.NR

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