ApCoCoA-1:BBSGen.TraceSyzFull

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} with s\in N^{+} and

    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

   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 that is equal to a non-standard degree of an element from \tau and 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