ApCoCoA-1:CharP.IMBBasisF2
CharP.IMBBasis
Computes 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
Introduction to Groebner Basis in CoCoA