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

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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.
 
<itemize>
 
<itemize>
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(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.</item>
+
<item>Before calling the function, please set ring environment coefficient field <tt>K</tt>, alphabet <tt>X</tt> and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(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.</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 <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>.  Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</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 [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>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item>
 
<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item>
 
<item>@return: a LIST which represents normal remainder of <tt>F</tt> with respect to <tt>Polynomials</tt>.</item>
 
<item>@return: a LIST which represents normal remainder of <tt>F</tt> with respect to <tt>Polynomials</tt>.</item>

Revision as of 21:17, 17 July 2010

NC.NR

Normal remainder polynomial with respect to a list of polynomials over a free associative K-algebra.

Syntax

NC.NR(F:LIST, Polynomials: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.

  • 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,""]].

  • @param Polynomials: a LIST of polynomials in K<X>.

  • @return: a LIST which represents normal remainder of F with respect to Polynomials.

Example

NC.RingEnv();
Coefficient ring : Q (float type in C++)
Alphabet : abc
Ordering : LLEX

-------------------------------
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; 
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
Polynomials:=[F1,F2];
NC.NR(F,Polynomials);
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]
-------------------------------
NC.SetOrdering(<quotes>ELIM</quotes>);
NC.NR(F,Polynomials);
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]
-------------------------------

See also

NC.Add

NC.GB

NC.IsGB

NC.LC

NC.LT

NC.LTIdeal

NC.Multiply

NC.SetFp

NC.SetOrdering

NC.SetRelations

NC.SetRules

NC.SetX

NC.Subtract

NC.UnsetFp

NC.UnsetOrdering

NC.UnsetRelations

NC.UnsetRules

NC.UnsetX

Gbmr.MRSubtract

Gbmr.MRMultiply

Gbmr.MRBP

Gbmr.MRIntersection

Gbmr.MRKernelOfHomomorphism

Gbmr.MRMinimalPolynomials

Introduction to CoCoAServer