Difference between revisions of "ApCoCoA-1:Weyl.InIw"
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(New page: <command> <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_descri...) |
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| − | <command> | + | {{Version|1}} |
| − | + | <command> | |
| − | + | <title>Weyl.InIw</title> | |
| + | <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> | + | 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 > 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> | + | <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> | + | <see>ApCoCoA-1:Weyl.Inw|Weyl.Inw</see> |
| − | <see>Weyl.WLT</see> | + | <see>ApCoCoA-1:Weyl.WLT|Weyl.WLT</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
<type>ideal</type> | <type>ideal</type> | ||
| − | |||
</types> | </types> | ||
<key>weyl.InIw</key> | <key>weyl.InIw</key> | ||
| − | <key> | + | <key>iniw</key> |
| − | <wiki-category>Package_weyl</wiki-category> | + | <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
