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 improved 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.IMBBasisF2(F);

    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=11
    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=11
    Total No. of Mutants are = 0
    The size of Matrix is:
    	No. of Rows=12
    	No. of Columns=15
    Total No. of Mutants are = 2
    The No. of Mutants of Minimum degree (Mutants used) are = 1
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=15
    Total No. of Mutants are = 2
    The No. of Mutants of Minimum degree (Mutants used) are = 1
    The size of Matrix is:
    	No. of Rows=16
    	No. of Columns=15
    Total No. of Mutants are = 2
    The No. of Mutants of Minimum degree (Mutants used) are = 1
    The size of Matrix is:
    	No. of Rows=17
    	No. of Columns=15
    Total No. of Mutants are = 1
    The No. of Mutants of Minimum degree (Mutants used) are = 1
    The size of Matrix is:
    	No. of Rows=17
    	No. of Columns=15
    Total No. of Mutants are = 2
    The No. of Mutants of Minimum degree (Mutants used) are = 2
    The size of Matrix is:
    	No. of Rows=18
    	No. of Columns=15
    Total No. of Mutants are = 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.IMBBasisF2(F,NSol);

    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=9
    The size of Matrix is:
    	No. of Rows=7
    	No. of Columns=14
    Total No. of Mutants are = 0
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=14
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=14
[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

CharP.MXLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.IMNLASolve

CharP.MBBasisF2