Difference between revisions of "ApCoCoA-1:BBSGen.TraceSyzStep"

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m (insert version info)
 
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{{Version|1}}
 
<command>
 
<command>
 
   <title>BBSGen.TraceSyzStep</title>
 
   <title>BBSGen.TraceSyzStep</title>
−
   <short_description>: This function only computes the  trace syzygy for the degree of the given monomial.  
+
   <short_description>This function computes the  trace polynomial T_{Pi,X}  with respect to a  given term Pi and a variable from ring K[x_1,...,x_N].(see <ref>ApCoCoA-1:BBSGen.TraceSyzFull|BBSGen.TraceSyzFull</ref>)
 
</short_description>
 
</short_description>
 
    
 
    
 
<syntax>
 
<syntax>
  
−
TraceSyzLin(OO,BO,N);
+
BBSGen.TraceSyzStep(Pi,X,OO,BO,N);
−
TraceSyzLin(OO:LIST,BO:LIST,N:INTEGER):LIST
+
BBSGen.TraceSyzStep(Pi:POLY,X:POLY,OO:LIST,BO:LIST,N:INTEGER):LIST
 
</syntax>
 
</syntax>
 
   <description>
 
   <description>
−
 
+
  Note the following:
−
Let  Tau^kl_ij :=t[k,l,i,j] be the (i,j) ^th entry of matrix the operation  [A_k,A_l]. The result of the Trace Syzygy computation is K[c]-linear combination of  Tau^kl_ij    that is equal to 0. This function only computes the  trace syzygy for the degree of the given monomial.  
+
  The chosen variable must be a divisor of the term Pi.
−
 
+
  Pi must be a product of at least two different indeterminates otherwise the result is 0.  
 +
 
  
 
<itemize>
 
<itemize>
−
   <item>@param  The Monomial Mon, the distinguished indterminate of choice,  order ideal OO, border BO, the number of Indeterminates of the Polynomial.
+
   <item>@param  The term Pi from K[x_1,...,x_N], the distinguished variable of choice from {x_1,...,x_N},  order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].
 
</item>
 
</item>
−
   <item>@return  Trace syzygy of the degree of the given monomial.</item>
+
   <item>@return  Trace polynomial T_{Pi,X} with respect to a term Pi and a variable X.</item>
 
</itemize>
 
</itemize>
  
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Use R::=QQ[x[1..2]];
 
Use R::=QQ[x[1..2]];
  
−
OO:=BB.Box([1,1]);
+
OO:=$apcocoa/borderbasis.Box([1,1]);
−
BO:=BB.Border(OO);
+
BO:=$apcocoa/borderbasis.Border(OO);
 
Mu:=Len(OO);
 
Mu:=Len(OO);
 
Nu:=Len(BO);
 
Nu:=Len(BO);
−
 
+
N:=Len(Indets());
−
Mon:=x[1]^2x[2];--------Target Monomial
+
Pi:=x[1]^2x[2];
  
 
X:=x[1];  ------------Choice of the Indeterminate
 
X:=x[1];  ------------Choice of the Indeterminate
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Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
 
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
 
   
 
   
−
  BBSGen.TraceSyzStep(Mon,X,OO,BO,N);
+
  BBSGen.TraceSyzStep(Pi,X,OO,BO,N);
 
    
 
    
−
  c[1,2]t[1,2,3,1] + c[2,2]t[1,2,3,2] + c[3,2]t[1,2,3,3] +
+
c[1,2]t[1,2,3,1] + c[2,2]t[1,2,3,2] +  
−
c[4,2]t[1,2,3,4] + c[1,4]t[1,2,4,1] + c[2,4]t[1,2,4,2] +  
+
c[3,2]t[1,2,3,3] + c[4,2]t[1,2,3,4] +  
−
c[3,4]t[1,2,4,3] + c[4,4]t[1,2,4,4] +
+
c[1,4]t[1,2,4,1] + c[2,4]t[1,2,4,2] +
 +
c[3,4]t[1,2,4,3] + c[4,4]t[1,2,4,4] +
 
  t[1,2,1,3] + t[1,2,2,4]
 
  t[1,2,1,3] + t[1,2,2,4]
  
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   </description>
 
   </description>
 
   <types>
 
   <types>
−
     <type>borderbasis</type>
+
     <type>bbsmingensys</type>
−
     <type>ideal</type>
+
     <type>poly</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
−
<see>BB.Border</see>
 
−
  <see>BB.Box</see>
 
−
<see>BBSGen.Wmat</see>
 
−
<see>BBSGen.TraceSyzLin</see>
 
−
<see>BBSGen.TraceSyzStepLin</see>
 
−
<see>BBSGen.TraceSyzFull</see>
 
  
−
   <key>Wmat</key>
+
<see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
−
   <key>BBSGen.Wmat</key>
+
<see>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</see>
−
   <key>bbsmingensys.Wmat</key>
+
<see>ApCoCoA-1:BBSGen.TraceSyzLinStep|BBSGen.TraceSyzLinStep</see>
−
   <wiki-category>Package_bbsmingensys</wiki-category>
+
<see>ApCoCoA-1:BBSGen.TraceSyzFull|BBSGen.TraceSyzFull</see>
 +
 
 +
   <key>TraceSyzStep</key>
 +
   <key>BBSGen.TraceSyzStep</key>
 +
   <key>bbsmingensys.TraceSyzStep</key>
 +
   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:52, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.TraceSyzStep

This function computes the trace polynomial T_{Pi,X} with respect to a given term Pi and a variable from ring K[x_1,...,x_N].(see BBSGen.TraceSyzFull)

Syntax

BBSGen.TraceSyzStep(Pi,X,OO,BO,N);
BBSGen.TraceSyzStep(Pi:POLY,X:POLY,OO:LIST,BO:LIST,N:INTEGER):LIST

Description

 Note the following:
 The chosen variable must be a divisor of the term Pi.
 Pi must be a product of at least two different indeterminates otherwise the result is 0.

  • @param The term Pi from K[x_1,...,x_N], the distinguished variable of choice from {x_1,...,x_N}, order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].

  • @return Trace polynomial T_{Pi,X} with respect to a term Pi and a variable X.


Example

Use R::=QQ[x[1..2]];

OO:=$apcocoa/borderbasis.Box([1,1]);
BO:=$apcocoa/borderbasis.Border(OO);
Mu:=Len(OO);
Nu:=Len(BO);
N:=Len(Indets());
Pi:=x[1]^2x[2];

X:=x[1];   ------------Choice of the Indeterminate

Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; 
 
 BBSGen.TraceSyzStep(Pi,X,OO,BO,N);
  
c[1,2]t[1,2,3,1] + c[2,2]t[1,2,3,2] + 
c[3,2]t[1,2,3,3] + c[4,2]t[1,2,3,4] + 
c[1,4]t[1,2,4,1] + c[2,4]t[1,2,4,2] +
 c[3,4]t[1,2,4,3] + c[4,4]t[1,2,4,4] +
 t[1,2,1,3] + t[1,2,2,4]

-------------------------------


BBSGen.Wmat

BBSGen.TraceSyzLin

BBSGen.TraceSyzLinStep

BBSGen.TraceSyzFull




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