Difference between revisions of "ApCoCoA-1:LinSyz.Resolution"

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{{Version|1}}
 
<command>  
 
<command>  
 
   <title>LinSyz.Resolution</title>
 
   <title>LinSyz.Resolution</title>
     <short_description>computing the Syzygy moduls for a Module, generated by linear forms</short_description>
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     <short_description>Computes syzygy modules of a module generated by linear forms.</short_description>
 
<syntax>
 
<syntax>
$linsyz.Resolution(Module,Int):List
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LinSyz.Resolution(M:MODULE,N:INT):LIST
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
This command computes the resolution of a given module, which is generated by vectors consisting of linear forms. Be aware of the fact that this is very new and may contain bugs! Also the linear forms may not have any constant component, so if your system has some, you have to make the system homogeneous first via introducing a new indeterminate.
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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.
 +
<par/>
 +
This command computes the resolution of a given module, which is generated by vectors consisting of linear forms. Be aware of the fact that this is not well tested and may contain bugs! Also the linear forms may not have any constant component, so if your system has some, you have to homogenize the system first via introducing a new indeterminate.
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 +
<itemize>
 +
<item>@param <em>M</em> A module.</item>
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<item>@param <em>N</em> A non-negative integer which gives the number of syzygy modules which will be computed .If <tt>N=0</tt> all syzygy modules will be computed, otherwise the first <tt>N</tt> syzygy modules will be computed.</item>
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<item>@return The first set in the resulting lists contains the '0-th' syzygy module and consists of the interreduced generators!</item>
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</itemize>
 +
 
 +
<example>
 +
Use P::=QQ[x,y,z];
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M:=Module([[x+y+z,x+y+z,x-y+z],[x-y,y-4z,x+2z],[x,y,z]]);
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BettiDiagram(M);
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        0
 +
----------
 +
1:    3
 +
----------
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Tot:    3
 +
-------------------------------
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LinSyz.Resolution(M,0);
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-- CoCoAServer: computing Cpu Time = 0
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-------------------------------
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[Vector(x + y + z, x + y + z, x - y + z), Vector(x - y, y - 4z, x + 2z), Vector(x, y, z)]
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-------------------------------
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LinSyz.Resolution(M,1);
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-- CoCoAServer: computing Cpu Time = 0
 +
-------------------------------
 +
[ ]
 +
-------------------------------
 +
</example>
  
If the second paramer is set to 0, all syzygy moduls will be computed. otherwise, if the second parameter is n, the first n syzygy moduls will be computed
 
 
     </description>
 
     </description>
 +
 
     <seealso>
 
     <seealso>
       <see>LinSyz.BettyNumber</see>
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       <see>ApCoCoA-1:LinSyz.BettyNumber|LinSyz.BettyNumber</see>
     <see>LinSyz.BettyNumbers</see>
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     <see>ApCoCoA-1:LinSyz.BettyNumbers|LinSyz.BettyNumbers</see>
 +
    <see>ApCoCoA-1:Introduction to Modules|Introduction to Modules</see>
 
   </seealso>
 
   </seealso>
     <key>heldt</key>
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     <key>linsyz.BettyNumber</key>
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<types>
     <wiki-category>Package_linsyz</wiki-category>
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<type>apcocoaserver</type>
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<type>module</type>
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</types>
 +
 
 +
     <key>Resolution</key>
 +
     <key>linsyz.Resolution</key>
 +
     <wiki-category>ApCoCoA-1:Package_linsyz</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 10:12, 7 October 2020

This article is about a function from ApCoCoA-1.

LinSyz.Resolution

Computes syzygy modules of a module generated by linear forms.

Syntax

LinSyz.Resolution(M:MODULE,N:INT):LIST

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 command computes the resolution of a given module, which is generated by vectors consisting of linear forms. Be aware of the fact that this is not well tested and may contain bugs! Also the linear forms may not have any constant component, so if your system has some, you have to homogenize the system first via introducing a new indeterminate.

  • @param M A module.

  • @param N A non-negative integer which gives the number of syzygy modules which will be computed .If N=0 all syzygy modules will be computed, otherwise the first N syzygy modules will be computed.

  • @return The first set in the resulting lists contains the '0-th' syzygy module and consists of the interreduced generators!

Example

Use P::=QQ[x,y,z];
M:=Module([[x+y+z,x+y+z,x-y+z],[x-y,y-4z,x+2z],[x,y,z]]);
BettiDiagram(M);
        0
----------
 1:     3
----------
Tot:    3
-------------------------------
LinSyz.Resolution(M,0);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[Vector(x + y + z, x + y + z, x - y + z), Vector(x - y, y - 4z, x + 2z), Vector(x, y, z)]
-------------------------------
LinSyz.Resolution(M,1);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[ ]
-------------------------------


See also

LinSyz.BettyNumber

LinSyz.BettyNumbers

Introduction to Modules