Difference between revisions of "ApCoCoA-1:Weyl.WNR"

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     <title>Weyl.WNR</title>
 
     <title>Weyl.WNR</title>
 
     <short_description>Computes normal remainder of a Weyl polynomial F with respect
 
     <short_description>Computes normal remainder of a Weyl polynomial F with respect
to a polynomial L or a set of polnomials in the list L. </short_description>
+
to a polynomial or a set of polnomials. </short_description>
 
<syntax>
 
<syntax>
Weyl.WNR(F:POLY,G:LIST/POLY):POLY
+
Weyl.WNR(F:POLY,G:POLY):POLY
 +
Weyl.WNR(F:POLY,G:LIST):POLY
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
Computes normal remainder of a Weyl polynomial F with respect to a polynomial L or a set of polynomials in the list L.
+
Computes the normal remainder of a Weyl polynomial F with respect to a polynomial G or a set of polynomials in the list G.
If L is Groebner basis then this function is used for ideal membership problem.
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If G is Groebner basis then this function is used for ideal membership problem.
  
'''F: ''' a Weyl polynomial in normal form.
+
<itemize>
 +
<item>@param <em>F</em> A Weyl polynomial in normal form.</item>
 +
<item>@param <em>G</em> A Weyl polynomial or a list of Weyl polynomials.</item>
 +
<item>@return The remainder as a weyl polynomial using normal remainder algorithm in Weyl algebra A_n.</item>
 +
</itemize>
  
'''L: ''' a Weyl polynomial or a list of Weyl polynomials.
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<em>Note:</em> All polynomials that are not in normal form should be first converted in to normal form using <ref>Weyl.WNormalForm</ref>, otherwise you may get unexpected results.
  
Output is a remainder R as a weyl polynomial using normal remainder algorithm in Weyl algebra A_n.
 
 
<example>
 
<example>
 
W3::=ZZ/(7)[x[1..3],d[1..3]];
 
W3::=ZZ/(7)[x[1..3],d[1..3]];
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-------------------------------
 
-------------------------------
 
</example>
 
</example>
<em>Note:</em> All polynomials that are not in normal form should be first converted in to normal form using Weyl.WNormalForm(L), otherwise you may get unexpected results.
+
 
 
   </description>
 
   </description>
 
     <seealso>
 
     <seealso>

Revision as of 12:39, 23 April 2009

Weyl.WNR

Computes normal remainder of a Weyl polynomial F with respect

to a polynomial or a set of polnomials.

Syntax

Weyl.WNR(F:POLY,G:POLY):POLY
Weyl.WNR(F:POLY,G:LIST):POLY

Description

Computes the normal remainder of a Weyl polynomial F with respect to a polynomial G or a set of polynomials in the list G.

If G is Groebner basis then this function is used for ideal membership problem.

  • @param F A Weyl polynomial in normal form.

  • @param G A Weyl polynomial or a list of Weyl polynomials.

  • @return The remainder as a weyl polynomial using normal remainder algorithm in Weyl algebra A_n.

Note: All polynomials that are not in normal form should be first converted in to normal form using Weyl.WNormalForm, otherwise you may get unexpected results.

Example

W3::=ZZ/(7)[x[1..3],d[1..3]];
Use W3;
F1:=-d[1]^3d[2]^5d[3]^5+x[2]^5;
F2:=-3x[2]d[2]^5d[3]^5+x[2]d[1]^3;
F3:=-2d[1]^4d[2]^5-x[1]d[2]^7+x[3]^3d[3]^5;
L:=[F1,F2,F3];
Weyl.WNR(F1,L);
0
-------------------------------
Weyl.WNR(F1,Gens(Ideal(F2,F3)));
-d[1]^3d[2]^5d[3]^5 + x[2]^5
-------------------------------
Weyl.WNR(x[2]^5-d[1]^3,L);
x[2]^5 - d[1]^3
-------------------------------
Weyl.WNR(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
-------------------------------


See also

Weyl.WNormalForm