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

From ApCoCoAWiki
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<seealso>
 
<seealso>
 
<see>NC.Add</see>
 
<see>NC.Add</see>
<see>NC.BP</see>
 
 
<see>NC.Deg</see>
 
<see>NC.Deg</see>
 
<see>NC.FindPolynomials</see>
 
<see>NC.FindPolynomials</see>
 
<see>NC.GB</see>
 
<see>NC.GB</see>
 +
<see>NC.HF</see>
 
<see>NC.Intersection</see>
 
<see>NC.Intersection</see>
 
<see>NC.IsGB</see>
 
<see>NC.IsGB</see>
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<see>NC.LC</see>
 
<see>NC.LC</see>
 
<see>NC.LT</see>
 
<see>NC.LT</see>
<see>NC.LTIdeal</see>
+
<see>NC.NR</see>
<see>NC.MRAdd</see>
+
<see>NC.ReducedGB</see>
 
<see>NC.MRBP</see>
 
<see>NC.MRBP</see>
<see>NC.MRIntersection</see>
 
<see>NC.MRKernelOfHomomorphism</see>
 
<see>NC.MRMinimalPolynomials</see>
 
<see>NC.MRMultiply</see>
 
<see>NC.MRReducedBP</see>
 
<see>NC.MRSubtract</see>
 
<see>NC.MinimalPolynomial</see>
 
<see>NC.Multiply</see>
 
<see>NC.ReducedBP</see>
 
<see>NC.ReducedGB</see>
 
 
<see>NC.SetFp</see>
 
<see>NC.SetFp</see>
 
<see>NC.SetOrdering</see>
 
<see>NC.SetOrdering</see>

Revision as of 23:20, 11 December 2010

NC.NR

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

Syntax

NC.NR(F:LIST, G: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 ring environment coefficient field K, alphabet (or indeterminates) X and ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), 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 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, xy^2x is represented as "xyyx", unit is represented as an empty string "". Then, polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. 0 polynomial is represented as an empty LIST [].

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

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

Example

NC.SetX(<quotes>abc</quotes>);
NC.RingEnv();
Coefficient ring : Q
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.Deg

NC.FindPolynomials

NC.GB

NC.HF

NC.Intersection

NC.IsGB

NC.KernelOfHomomorphism

NC.LC

NC.LT

NC.NR

NC.ReducedGB

NC.MRBP

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