ApCoCoA-1:CharP.MBBasisF2
CharP.MBBasis
Computing a Border Basis of a given ideal over F_2.
Syntax
CharP.MBBasisF2(F:LIST):LIST CharP.MBBasisF2(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
