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

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   {{Version|1}}
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     <title>Weyl.AnnFs</title>
 
     <title>Weyl.AnnFs</title>
     <short_description>Computes annihilating ideal of a polynomial F^s in Weyl algebra <tt>A_n</tt>.</short_description>
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     <short_description>Computes annihilating ideal of a polynomial <tt>F^s</tt> in Weyl algebra <tt>A_n</tt>.</short_description>
 
<syntax>
 
<syntax>
 
Weyl.AnnFs(F:POLY):IDEAL
 
Weyl.AnnFs(F:POLY):IDEAL
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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.
 
<par/>
 
<par/>
This function computes annihilating ideal of a polynomial F^s using the Algorithm of Oaku and Takayama, where F is a polynomial in Weyl algebra D. F should not involve any of the indeterminates in {y1, ..., yn}. This ideal belongs to the Weyl algebra A_s =D[s]= QQ[x1, ..., xn, y1, ..., yn, s,w] where s commutes with all x_i and y_i's and w is redundant indeterminate used just to create internal structure of the weyl algebra.
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This function computes annihilating ideal of a polynomial <tt>F^s</tt> using the Algorithm of Oaku and Takayama, where <tt>F</tt> is a polynomial in Weyl algebra <tt>A_n</tt>. <tt>F</tt> should not involve any of the indeterminates in <tt>{y1, ..., yn}</tt>. This ideal belongs to the Weyl algebra <tt>A_s =D[s]= QQ[x1, ..., xn, y1, ..., yn, s,w]</tt> where <tt>s</tt> commutes with all <tt>x_i</tt> and <tt>y_i</tt>'s and <tt>w</tt> is redundant indeterminate used just to create internal structure of the weyl algebra.
  
 
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<item>@param <em>F</em> A polynomial F in the indeterminates x1, ..., xn of a Weyl Algebra D.</item>
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<item>@param <em>F</em> A polynomial <tt>F</tt> in the indeterminates <tt>x1, ..., xn</tt> of a Weyl Algebra <tt>A_n</tt>.</item>
<item>@return An ideal in A_s=QQ[x1, ..., xn,y1, ...,yn, s,w].</item>
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<item>@return An ideal in <tt>A_s=QQ[x1, ..., xn,y1, ...,yn, s,w]</tt>.</item>
 
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</itemize>
  
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   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
       <see>Introduction to CoCoAServer</see>
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       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
     </seealso>
 
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     </types>
 
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     <key>weyl.annfs</key>
 
     <key>weyl.annfs</key>
     <key>chari</key>
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     <key>annfs</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 10:34, 7 October 2020

This article is about a function from ApCoCoA-1.

Weyl.AnnFs

Computes annihilating ideal of a polynomial F^s in Weyl algebra A_n.

Syntax

Weyl.AnnFs(F:POLY):IDEAL

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.

This function computes annihilating ideal of a polynomial F^s using the Algorithm of Oaku and Takayama, where F is a polynomial in Weyl algebra A_n. F should not involve any of the indeterminates in {y1, ..., yn}. This ideal belongs to the Weyl algebra A_s =D[s]= QQ[x1, ..., xn, y1, ..., yn, s,w] where s commutes with all x_i and y_i's and w is redundant indeterminate used just to create internal structure of the weyl algebra.

  • @param F A polynomial F in the indeterminates x1, ..., xn of a Weyl Algebra A_n.

  • @return An ideal in A_s=QQ[x1, ..., xn,y1, ...,yn, s,w].

Example

A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring
Use A2;
F:=x[1]^3-x[2]^2;
-------------------------------
AnnI:=Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.078
-------------------------------
Ideal of Ring A_s = QQ[x[1..2],y[1..2],s,w]
Where current indeterminates are mapped into ring A_s as follows:
x[1] --> x[1] and d[1] --> y[1]
x[2] --> x[2] and d[2] --> y[2]

-------------------------------
AnnI;
A_s :: Ideal(
  3x[1]^2y[2] + 2x[2]y[1],
  2x[1]y[1] + 3x[2]y[2] - 6s)   --AnnI belongs to the new ring A_s
-------------------------------

Example

A3::=QQ[x[1..3],d[1..3]]; --Define appropriate ring
Use A3;

F:=x[2]^2-x[1]x[3]-1;
Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.14
-------------------------------
Ideal of Ring A_s = QQ[x[1..3],y[1..3],s,w]
A_s :: Ideal(2x[2]y[1] + x[3]y[2], x[1]y[1] - x[3]y[3], x[2]x[3]y[2] + 2x[3]^2y[3] - 2x[3]s + 2y[1], x[2]^2y[2] + 
2x[2]x[3]y[3] - 2x[2]s - y[2], -x[2]^2y[3] + x[1]x[3]y[3] - x[1]s + y[3], x[1]y[2] + 2x[2]y[3])
-------------------------------

See also

Introduction to CoCoAServer