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

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   {{Version|1}}
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<command>
 
<title>Weyl.InIw</title>
 
<title>Weyl.InIw</title>
<short_description>Computes the initial ideal of a D-ideal I in Weyl algebra A_n with respect to the weight vector W=(u_i,v_i).</short_description>
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<short_description>Computes the initial ideal of a D-ideal <tt>I</tt> in Weyl algebra <tt>A_n</tt> with respect to the weight vector <tt>W=(u_i,v_i)</tt>.</short_description>
 
<syntax>
 
<syntax>
 
Weyl.InIw(I:IDEAL,W:LIST):IDEAL
 
Weyl.InIw(I:IDEAL,W:LIST):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/>
Computes the initial ideal of a D-ideal I in the Weyl algebra D with respect to weight vector W:=[u,v] such that u+v>0. Here u=(u1,...,un) and v=(v1,...,vn) are weights of indeterminates [x1,...,xn] and [y1,...,yn] respectively. Note that InIw(P,W) is an ideal of graded ring of D with respect to weight vector W. Due to limitations in CoCoA4, all u_i and v_i should be positive integers. Computation of initial ideal when u+v= 0 is not implemented yet.
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Computes the initial ideal of a D-ideal <tt>I</tt> in the Weyl algebra D with respect to weight vector <tt>W:=[u,v]</tt> such that <tt>u+v &gt; 0</tt>. Here <tt>u=(u1,...,un)</tt> and <tt>v=(v1,...,vn)</tt> are weights of indeterminates <tt>[x1,...,xn]</tt> and <tt>[y1,...,yn]</tt> respectively. Note that <tt>InIw(P,W)</tt> is an ideal of graded ring of D with respect to weight vector <tt>W</tt>. Due to limitations in CoCoA4, all <tt>u_i</tt> and <tt>v_i</tt> should be positive integers. Computation of initial ideal if <tt>u+v= 0</tt> is not implemented yet.
 
<itemize>
 
<itemize>
 
<item>@param <em>I</em> An ideal in the Weyl algebra.</item>
 
<item>@param <em>I</em> An ideal in the Weyl algebra.</item>
 
<item>@param <em>W</em> A list of n positive integers, where n = number of indeterminates.</item>
 
<item>@param <em>W</em> A list of n positive integers, where n = number of indeterminates.</item>
<item>@return An ideal, which is the initial ideal of I with respect to W.</item>
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<item>@return An ideal, which is the initial ideal of <tt>I</tt> with respect to <tt>W</tt>.</item>
 
</itemize>
 
</itemize>
 
<em>Beta Warning:</em> This method, package or class is a beta version. It may not work as intended or its interface may change in the next version! So please be careful when you're intending to use it.
 
<em>Beta Warning:</em> This method, package or class is a beta version. It may not work as intended or its interface may change in the next version! So please be careful when you're intending to use it.
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   </description>
 
   </description>
 
     <seealso>
 
     <seealso>
       <see>Weyl.Inw</see>
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       <see>ApCoCoA-1:Weyl.Inw|Weyl.Inw</see>
       <see>Weyl.WLT</see>
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       <see>ApCoCoA-1:Weyl.WLT|Weyl.WLT</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
 
       <type>apcocoaserver</type>
 
       <type>apcocoaserver</type>
 
       <type>ideal</type>
 
       <type>ideal</type>
      <type>groebner</type>
 
 
     </types>
 
     </types>
 
     <key>weyl.InIw</key>
 
     <key>weyl.InIw</key>
     <key>wgb</key>
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     <key>iniw</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 13:50, 29 October 2020

This article is about a function from ApCoCoA-1.

Weyl.InIw

Computes the initial ideal of a D-ideal I in Weyl algebra A_n with respect to the weight vector W=(u_i,v_i).

Syntax

Weyl.InIw(I:IDEAL,W:LIST):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.

Computes the initial ideal of a D-ideal I in the Weyl algebra D with respect to weight vector W:=[u,v] such that u+v > 0. Here u=(u1,...,un) and v=(v1,...,vn) are weights of indeterminates [x1,...,xn] and [y1,...,yn] respectively. Note that InIw(P,W) is an ideal of graded ring of D with respect to weight vector W. Due to limitations in CoCoA4, all u_i and v_i should be positive integers. Computation of initial ideal if u+v= 0 is not implemented yet.

  • @param I An ideal in the Weyl algebra.

  • @param W A list of n positive integers, where n = number of indeterminates.

  • @return An ideal, which is the initial ideal of I with respect to W.

Beta Warning: This method, package or class is a beta version. It may not work as intended or its interface may change in the next version! So please be careful when you're intending to use it.

Example

Use A4::=QQ[x[1..4],d[1..4]];
I4:=Ideal(d[2]d[3]-d[1]d[4],x[1]d[1]-x[4]d[4]-1,x[2]d[2]+x[4]d[4]+1,x[3]d[3]+x[4]d[4]+2);
Weyl.InIw(I4,[0,0,0,0,1,1,1,1]);  --This function is implemented only for positive weights!
ERROR: All weights shoud be strictly positive
CONTEXT: Error("All weights shoud be strictly positive")
-------------------------------
Weyl.InIw(I4,[1,1,1,1,2,2,1,1]);
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
Ideal(x[2]x[3]x[4]d[4]^2 - x[1]x[4]^2d[4]^2, x[1]x[4]d[3]d[4] + x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4], x[2]d[1]d[4],
x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1], x[2]d[2], x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------
Weyl.InIw(I4,[1,2,1,1,2,2,1,1]);
-- CoCoAServer: computing Cpu Time = 0.032
-------------------------------
Ideal(x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4], x[2]d[1]d[4], x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1], x[2]d[2],
x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------

Weyl.InIw(I4,[2,2,2,2,1,1,1,1]);
-- CoCoAServer: computing Cpu Time = 0.031
-------------------------------
Ideal(x[2]x[3]x[4]d[4]^2 - x[1]x[4]^2d[4]^2, x[1]x[4]d[3]d[4] + x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4] + x[3]x[4]d[4]^2,
x[2]d[1]d[4] + x[4]d[3]d[4], x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1] - x[4]d[4], x[2]d[2] + x[4]d[4],
x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------


See also

Weyl.Inw

Weyl.WLT