Difference between revisions of "ApCoCoA-1:Weyl.AnnFs"
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| − | <command> | + | {{Version|1}} |
| + | <command> | ||
<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> | + | <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 | + | 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. |
<itemize> | <itemize> | ||
| − | <item>@param <em>F</em> A polynomial F in the indeterminates x1, ..., xn of a Weyl Algebra | + | <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> | + | <item>@return An ideal in <tt>A_s=QQ[x1, ..., xn,y1, ...,yn, s,w]</tt>.</item> |
</itemize> | </itemize> | ||
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------------------------------- | ------------------------------- | ||
Ideal of Ring A_s = QQ[x[1..3],y[1..3],s,w] | 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]) | + | 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]) | ||
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
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</types> | </types> | ||
<key>weyl.annfs</key> | <key>weyl.annfs</key> | ||
| − | <key> | + | <key>annfs</key> |
| − | <wiki-category>Package_weyl</wiki-category> | + | <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
