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

From ApCoCoAWiki
m (insert version info)
 
(15 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 +
{{Version|1}}
 
<command>
 
<command>
 
   <title>BBSGen.TraceSyzFull</title>
 
   <title>BBSGen.TraceSyzFull</title>
   <short_description>: This function computes the trace syzygy polynomials.
+
   <short_description>This function computes the trace polynomials.
  
 
              
 
              
Line 11: Line 12:
 
</syntax>
 
</syntax>
 
   <description>
 
   <description>
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 computes such Trace Syzygy polynomials.  This function, because of the growth of the polynomials during the computation,may not give result for every ring with three indeterminates. In that case we recommend the functions TraceSyzLin or TraceSyzStep.
+
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}.   
 +
<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 <ref>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</ref> and
 +
<ref>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</ref> .
 +
 
 +
 
  
  
 
<itemize>
 
<itemize>
   <item>@param The order ideal OO, border BO, the number of Indeterminates of the Polynomial.
+
   <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>
Line 25: Line 37:
 
      
 
      
 
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);
W:=BBSGen.Wmat(OO,BO,N);
+
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]];  
Line 57: Line 70:
 
   </description>
 
   </description>
 
   <types>
 
   <types>
     <type>borderbasis</type>
+
     <type>bbsmingensys</type>
     <type>ideal</type>
+
     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
<see>BB.Border</see>
 
  <see>BB.Box</see>
 
<see>BBSGen.Wmat</see>
 
<see>BBSGen.TraceSyzStep</see>
 
<see>BBSGen.TraceSyzLin</see>
 
  
   <key>Wmat</key>
+
<see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
   <key>BBSGen.Wmat</key>
+
<see>ApCoCoA-1:BBSGen.TraceSyzStep|BBSGen.TraceSyzStep</see>
   <key>bbsmingensys.Wmat</key>
+
<see>ApCoCoA-1:BBSGen.TraceSyzLin|BBSGen.TraceSyzLin</see>
   <wiki-category>Package_bbsmingensys</wiki-category>
+
 
 +
   <key>TraceSyzFull</key>
 +
   <key>BBSGen.TraceSyzFull</key>
 +
   <key>bbsmingensys.TraceSyzFull</key>
 +
   <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