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

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   <command>
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   {{Version|1}}
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<command>
 
     <title>Weyl.WNR</title>
 
     <title>Weyl.WNR</title>
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     <short_description>Computes normal remainder of a Weyl polynomial F with respect
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     <short_description>Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect
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to a polynomial L or a set of polnomials in the list L. </short_description>
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to a polynomial or a list of Weyl polynomials using corresponding implementation in ApCoCoALib.</short_description>
 
<syntax>
 
<syntax>
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Weyl.WNR(F:POLY,G:LIST/POLY):POLY
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Weyl.WNR(F:POLY,G:POLY):POLY
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Weyl.WNR(F:POLY,G:LIST):POLY
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
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Computes normal remainder of a Weyl polynomial F with respect to a polynomial L or a set of polynomials in the list L.
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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.
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If L is Groebner basis then this function is used for ideal membership problem.
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<par/>
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Computes the normal remainder of a Weyl polynomial <tt>F</tt> with respect to a polynomial <tt>G</tt> or a set of polynomials in the list <tt>G</tt>.
 +
If <tt>G</tt> is Groebner basis then this function is used for ideal membership problem. That is, a Weyl polynomial P belongs to an ideal I iff <tt>Weyl.WNR(P,Weyl.WGB(I))=0</tt>.
  
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'''F: ''' a Weyl polynomial in normal form.
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<itemize>
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<item>@param <em>F</em> A Weyl polynomial in normal form.</item>
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<item>@param <em>G</em> A Weyl polynomial or a list of Weyl polynomials.</item>
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<item>@return The remainder as a Weyl polynomial using normal remainder algorithm in Weyl algebra <tt>A_n</tt>.</item>
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</itemize>
  
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'''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 into normal form using <ref>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</ref>, otherwise you may get unexpected results.
  
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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]];
Line 23: Line 30:
 
L:=[F1,F2,F3];
 
L:=[F1,F2,F3];
 
Weyl.WNR(F1,L);
 
Weyl.WNR(F1,L);
 +
-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 
0
 
0
 +
-------------------------------
 +
-- Done.
 
-------------------------------
 
-------------------------------
 
Weyl.WNR(F1,Gens(Ideal(F2,F3)));
 
Weyl.WNR(F1,Gens(Ideal(F2,F3)));
 +
-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 
-d[1]^3d[2]^5d[3]^5 + x[2]^5
 
-d[1]^3d[2]^5d[3]^5 + x[2]^5
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-------------------------------
 +
-- Done.
 
-------------------------------
 
-------------------------------
 
Weyl.WNR(x[2]^5-d[1]^3,L);
 
Weyl.WNR(x[2]^5-d[1]^3,L);
 +
-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 
x[2]^5 - d[1]^3
 
x[2]^5 - d[1]^3
 +
-------------------------------
 +
-- Done.
 
-------------------------------
 
-------------------------------
 
Weyl.WNR(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);
 
Weyl.WNR(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);
 +
 +
-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
 
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
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-------------------------------
 +
-- Done.
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
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<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.
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<example>
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Use A1::=QQ[x,d];
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Weyl.WNR(xd,d);
 +
 
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 +
0
 +
-------------------------------
 +
-- Done.
 +
-------------------------------
 +
Weyl.WNR(xd,x);
 +
 
 +
-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 +
-1
 +
-------------------------------
 +
-- Done.
 +
-------------------------------
 +
</example>
 +
 
 
   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
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       <see>Weyl.WNormalForm</see>
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       <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see>
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      <see>ApCoCoA-1:Weyl.WNormalRemainder|Weyl.WNormalRemainder</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
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       <type>cocoaserver</type>
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       <type>apcocoaserver</type>
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      <type>polynomial</type>
 
     </types>
 
     </types>
 
     <key>weyl.wnr</key>
 
     <key>weyl.wnr</key>
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     <wiki-category>Package_weyl</wiki-category>
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    <key>Weyl.WNR</key>
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    <key>WNR</key>
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     <wiki-category>ApCoCoA-1:Package_weyl</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 10:39, 7 October 2020

This article is about a function from ApCoCoA-1.

Weyl.WNR

Computes the normal remainder of a Weyl polynomial F with respect

to a polynomial or a list of Weyl polynomials using corresponding implementation in ApCoCoALib.

Syntax

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

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.

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. That is, a Weyl polynomial P belongs to an ideal I iff Weyl.WNR(P,Weyl.WGB(I))=0.

  • @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 into 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);
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
0
-------------------------------
-- Done.
-------------------------------
Weyl.WNR(F1,Gens(Ideal(F2,F3)));
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
-d[1]^3d[2]^5d[3]^5 + x[2]^5
-------------------------------
-- Done.
-------------------------------
Weyl.WNR(x[2]^5-d[1]^3,L);
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
x[2]^5 - d[1]^3
-------------------------------
-- Done.
-------------------------------
Weyl.WNR(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);

-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
-------------------------------
-- Done.
-------------------------------

Example

Use A1::=QQ[x,d];
Weyl.WNR(xd,d);

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
0
-------------------------------
-- Done.
-------------------------------
Weyl.WNR(xd,x);

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
-1
-------------------------------
-- Done.
-------------------------------


See also

Weyl.WNormalForm

Weyl.WNormalRemainder




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