# ApCoCoA-1:CharP.IMBBasisF2

## CharP.IMBBasis

Computing a Border Basis of a given ideal over `F_2`.

### Syntax

CharP.IMBBasisF2(F:LIST):LIST CharP.IMBBasisF2(F:LIST, NSol: INT):LIST

### Description

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Let `f_1`, ... , `f_m` is a set of polynomials which generate a zero-dimensional ideal. This function computes a Border Basis of the zero-dimensional radical ideal generated by `f_1`, ... , `f_m` and the field polynomials. Furthermore, it uses mutant strategy to compute a U-stable span. If you want to use the second version with the parameter NSol, you need to provide the exact number of `F_2` rational solutions. The first version is safe to use if you do not know the exact number of `F_2` rational solutions.

@param

*F:*List of polynomials.@param

*NSol:*Number of`F_2`rational solutions.@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials.

#### Example

Use Z/(2)[x[1..4]]; F:=[ x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1 ]; -- Then we compute a Border Basis with CharP.MBBasisF2(F); The size of Matrix is: No. of Rows=4 No. of Columns=11 The size of Matrix is: No. of Rows=8 No. of Columns=11 No. of mutants found =1 The size of Matrix is: No. of Rows=11 No. of Columns=11 No. of mutants found =2 The size of Matrix is: No. of Rows=16 No. of Columns=11 No. of mutants found =0 The size of Matrix is: No. of Rows=31 No. of Columns=15 No. of mutants found =0 [x[4] + 1, x[3], x[2] + 1, x[1]]

#### Example

Use Z/(2)[x[1..4]]; F:=[ x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4], x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2], x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2] ]; NSol:=3; -- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions -- Compute the solution with CharP.MBBasisF2(F,NSol); The size of Matrix is: No. of Rows=4 No. of Columns=9 The size of Matrix is: No. of Rows=14 No. of Columns=14 The size of Matrix is: No. of Rows=16 No. of Columns=15 [x[3]x[4] + x[4], x[1]x[4] + x[1], x[1]x[3] + x[1], x[1]x[2] + x[1], x[2]x[3]x[4] + x[4], x[1]x[2]x[4] + x[1]]

### See also

Introduction to Groebner Basis in CoCoA