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

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   <command>
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   {{Version|1}}
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<command>
 
     <title>Weyl.WRGB</title>
 
     <title>Weyl.WRGB</title>
 
     <short_description>Reduced Groebner basis of an ideal <tt>I</tt> in Weyl algebra <tt>A_n</tt>.</short_description>
 
     <short_description>Reduced Groebner basis of an ideal <tt>I</tt> in Weyl algebra <tt>A_n</tt>.</short_description>
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     <description>
 
     <description>
 
<par/>
 
<par/>
This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>Weyl.WNR</ref> into a new list <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1.
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This function converts a Weyl Groebner basis <tt>GB</tt> computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If <tt>GB</tt> is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list <tt>GB</tt> of Weyl polynomials using <ref>ApCoCoA-1:Weyl.WNR|Weyl.WNR</ref> into a new list <tt>L</tt> such that <tt>Ideal(L) = Ideal(GB)</tt>, every polynomial is reduced with respect to the remaining polynomials in the list <tt>L</tt> and leading coefficient of each polynomial in <tt>L</tt> is 1.
  
 
<itemize>
 
<itemize>
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Weyl.WRGB(L);
 
Weyl.WRGB(L);
 
[1]
 
[1]
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-------------------------------
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</example>
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<example>
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A2::=ZZ/7[x[1..2],y[1..2]]; -- define appropriate ring
 +
Use A2;
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I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1);
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GbI:=Weyl.WGB(I,0);Len(GbI);
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 +
-------------------------------
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-- CoCoAServer: computing Cpu Time = 0.485
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-------------------------------
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42  -- size of complete GB of the ideal I
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-------------------------------
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Time GbI:=Weyl.WRGB(GbI);Len(GbI);
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Cpu time = 9.61, User time = 10
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-------------------------------
 +
11
 +
-------------------------------
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-- Done.
 +
-------------------------------
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Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib
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                                    -- note that this speeds up the computations
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-------------------------------
 +
-- CoCoAServer: computing Cpu Time = 0 
 +
-------------------------------
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Cpu time = 0.04, User time = 0 
 +
-------------------------------
 +
11  -- this is now size of reduced GB of the ideal I
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-------------------------------
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-- Done.
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
       <see>Weyl.WNormalForm</see>
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       <see>ApCoCoA-1:Weyl.WNormalForm|Weyl.WNormalForm</see>
       <see>Weyl.WGB</see>
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       <see>ApCoCoA-1:Weyl.WGB|Weyl.WGB</see>
       <see>Weyl.WRGBS</see>
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       <see>ApCoCoA-1:Weyl.WRGBS|Weyl.WRGBS</see>
       <see>Weyl.WRedGB</see>
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       <see>ApCoCoA-1:Weyl.WRedGB|Weyl.WRedGB</see>
       <see>Introduction to Groebner Basis in CoCoA</see>
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       <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
       <see>Introduction to CoCoAServer</see>
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       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
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     <key>Weyl.WRGB</key>
 
     <key>Weyl.WRGB</key>
 
     <key>wrgb</key>
 
     <key>wrgb</key>
     <wiki-category>Package_weyl</wiki-category>
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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.WRGB

Reduced Groebner basis of an ideal I in Weyl algebra A_n.

Syntax

Weyl.WRGB(GB:LIST):LIST

Description


This function converts a Weyl Groebner basis GB computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If GB is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list GB of Weyl polynomials using Weyl.WNR into a new list L such that Ideal(L) = Ideal(GB), every polynomial is reduced with respect to the remaining polynomials in the list L and leading coefficient of each polynomial in L is 1.

  • @param GB Groebner Basis of an ideal in the Weyl algebra.

  • @result The reduced Groebner Basis of the given ideal.

Example

A1::=QQ[x,d];	--Define appropriate ring
Use A1;
L:=[x,d,1];
Weyl.WRGB(L);
[1]
-------------------------------

Example

A2::=ZZ/7[x[1..2],y[1..2]]; -- define appropriate ring
Use A2;
I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1);
GbI:=Weyl.WGB(I,0);Len(GbI);

-------------------------------
-- CoCoAServer: computing Cpu Time = 0.485
-------------------------------
42   -- size of complete GB of the ideal I
-------------------------------
Time GbI:=Weyl.WRGB(GbI);Len(GbI);

Cpu time = 9.61, User time = 10
-------------------------------
11
-------------------------------
-- Done.
-------------------------------
Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib
                                    -- note that this speeds up the computations
-------------------------------
-- CoCoAServer: computing Cpu Time = 0  
-------------------------------
Cpu time = 0.04, User time = 0   
-------------------------------
11   -- this is now size of reduced GB of the ideal I
-------------------------------
-- Done.
-------------------------------

See also

Weyl.WNormalForm

Weyl.WGB

Weyl.WRGBS

Weyl.WRedGB

Introduction to Groebner Basis in CoCoA

Introduction to CoCoAServer