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

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{{Version|1}}
 
<command>
 
<command>
 
   <title>BBSGen.TraceSyzFull</title>
 
   <title>BBSGen.TraceSyzFull</title>
   <short_description>: This function computes the trace polynomials.
+
   <short_description>This function computes the trace polynomials.
  
 
              
 
              
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</syntax>
 
</syntax>
 
   <description>
 
   <description>
Let l,k_{1},....,k_{s}\in {1,...,n} with s\in N^{+} and
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Let l,k_1,....,k_s in {1,...,n} where s is a positive  integer and
     Pi=x_{k_{1}}...x_{k_{s}}x_{l}
+
     Pi=x_{k_1}...x_{k_s}x_l
a term (or power product) from the given Ring K[x_1,...,x_N]. Let the generic multiplication matrices A_{k_{1}},...,A_{k_{s}},A_{l}\in Mat(K[c]) be  associated to the indeterminates in \Pi. We shall name the polynomial   
+
a term (or power product) from the polynomial ring K[x_1,...,x_N]. Let the generic multiplication matrices A_{k_1},...,A_{k_s},A_l in Mat(K[c]) be  associated to the indeterminates in Pi. We shall name the polynomial   
     Trace([A_{k_{1}}...A_{k_{s}},A_{l}])\in K[c]  
+
     Trace([A_{k_1}...A_{k_s},A_l]) in K[c]  
as the trace  polynomial with respect to  Pi  and variable  x_{l}. We shall denote it by   
+
as the trace  polynomial with respect to  Pi  and variable  x_l. We shall denote it by   
   T_{Pi,x_{l}}}.   
+
   T_{Pi,x_l}.   
This function computes every trace polynomial with respect to every Pi that is equal to a non-standard degree of an element from \tau and every variable from {x_1,...,x_N}.   
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This function computes every trace polynomial with respect to every Pi with log that is equal to a non-standard degree of an element from tau and with respect to every variable from {x_1,...,x_N}.   
 
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<par/>
NOTE: This function due to the growth of polynomials during the matrix multiplication, may not give result for every ring and order ideal. In that case we recommend <see>BBSGen.TraceSyzStep</see> and
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NOTE: This function due to the growth of polynomials during the matrix multiplication, may not give result for every ring and order ideal. In that case we recommend <ref>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</ref> and
<see>BBSGen.TraceSyzLin</see> .
+
<ref>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</ref> .
  
  
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<itemize>
 
<itemize>
   <item>@param The order ideal OO, border BO, the number of Indeterminates of the Polynomial.(see <commandref>BB.Box</commandref>, <commandref>BB.Border</commandref> in package borderbasis)
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   <item>@param The order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].
 
</item>
 
</item>
 
   <item>@return  The list of Trace Syzygy polynomials.  </item>
 
   <item>@return  The list of Trace Syzygy polynomials.  </item>
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Use R::=QQ[x[1..2]];
 
Use R::=QQ[x[1..2]];
OO:=BB.Box([1,1]);
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OO:=$apcocoa/borderbasis.Box([1,1]);
BO:=BB.Border(OO);
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BO:=$apcocoa/borderbasis.Border(OO);
W:=BBSGen.Wmat(OO,BO,N);
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N:=Len(Indets());
 +
W:=BBSGen.Wmat(OO,BO,N);
 
Mu:=Len(OO);
 
Mu:=Len(OO);
 
Nu:=Len(BO);
 
Nu:=Len(BO);
N:=Len(Indets());
+
 
  
 
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]];  
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   </description>
 
   </description>
 
   <types>
 
   <types>
     <type>borderbasis</type>
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     <type>bbsmingensys</type>
     <type>ideal</type>
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     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
  
  <see>BBSGen.Wmat</see>
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  <see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
<see>BBSGen.TraceSyzStep</see>
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<see>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</see>
<see>BBSGen.TraceSyzLin</see>
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<see>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</see>
  
   <key>Wmat</key>
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   <key>TraceSyzFull</key>
   <key>BBSGen.Wmat</key>
+
   <key>BBSGen.TraceSyzFull</key>
   <key>bbsmingensys.Wmat</key>
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   <key>bbsmingensys.TraceSyzFull</key>
   <wiki-category>Package_bbsmingensys</wiki-category>
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   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:51, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.TraceSyzFull

This function computes the trace polynomials.


Syntax

TraceSyzFull(OO,BO,N);
TraceSyzFull(OO:LIST,BO:LIST,N:INTEGER):LIST

Description

Let l,k_1,....,k_s in {1,...,n} where s is a positive integer and

    Pi=x_{k_1}...x_{k_s}x_l

a term (or power product) from the polynomial ring K[x_1,...,x_N]. Let the generic multiplication matrices A_{k_1},...,A_{k_s},A_l in Mat(K[c]) be associated to the indeterminates in Pi. We shall name the polynomial

   Trace([A_{k_1}...A_{k_s},A_l]) in K[c] 

as the trace polynomial with respect to Pi and variable x_l. We shall denote it by

 T_{Pi,x_l}.  

This function computes every trace polynomial with respect to every Pi with log that is equal to a non-standard degree of an element from tau and with respect to every variable from {x_1,...,x_N}.

NOTE: This function due to the growth of polynomials during the matrix multiplication, may not give result for every ring and order ideal. In that case we recommend BBSGen.TraceSyzStep and

BBSGen.TraceSyzLin .



  • @param The order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].

  • @return The list of Trace Syzygy polynomials.


Example

    
Use R::=QQ[x[1..2]];
OO:=$apcocoa/borderbasis.Box([1,1]);
BO:=$apcocoa/borderbasis.Border(OO);
N:=Len(Indets());
W:=BBSGen.Wmat(OO,BO,N);
Mu:=Len(OO);
Nu:=Len(BO);


Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; 


BBSGen.TraceSyzFull(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],
  2c[1,1]t[1,2,2,1] + 2c[2,1]t[1,2,2,2] + 2c[3,1]t[1,2,2,3] + 2c[4,1]t[1,2,2,4]+ 
2c[1,3]t[1,2,4,1] + 2c[2,3]t[1,2,4,2] + 2c[3,3]t[1,2,4,3] + 2c[4,3]t[1,2,4,4] 
+ 2t[1,2,1,2] + 2t[1,2,3,4],
  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],
  2c[1,2]c[3,1]t[1,2,2,1] + 2c[1,4]c[4,1]t[1,2,2,1] + 2c[2,2]c[3,1]t[1,2,2,2] +
 2c[2,4]c[4,1]t[1,2,2,2] + 2c[3,1]c[3,2]t[1,2,2,3] + 2c[3,4]c[4,1]t[1,2,2,3] +
 2c[3,1]c[4,2]t[1,2,2,4] + 2c[4,1]c[4,4]t[1,2,2,4] + 2c[1,2]c[3,3]t[1,2,4,1] +
 2c[1,4]c[4,3]t[1,2,4,1] + 2c[2,2]c[3,3]t[1,2,4,2] + 2c[2,4]c[4,3]t[1,2,4,2] +
 2c[3,2]c[3,3]t[1,2,4,3] + 2c[3,4]c[4,3]t[1,2,4,3] + 2c[3,3]c[4,2]t[1,2,4,4] + 
2c[4,3]c[4,4]t[1,2,4,4] + 2c[1,1]t[1,2,2,3] + 2c[2,1]t[1,2,2,4] + 2c[1,4]t[1,2,3,1] + 
2c[2,4]t[1,2,3,2] + 2c[3,4]t[1,2,3,3] + 2c[4,4]t[1,2,3,4] + 2c[1,3]t[1,2,4,3] +
 2c[2,3]t[1,2,4,4] + 2t[1,2,1,4]]


BBSGen.Wmat

BBSGen.TraceSyzStep

BBSGen.TraceSyzLin