# 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} 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

• @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]]
```