ApCoCoA-1:CharP.MBBasisF2

CharP.MBBasis

Computing a Border Basis of a given ideal over F_2.

Syntax

CharP.MBBasisF2(F:LIST):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.

• @param F: List of polynomials.

• @return A Border Basis of 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 the solution with
CharP.MXLSolve(F);

-- And we achieve the following information on the screen together with the solution at the end.
----------------------------------------
The size of Matrix is:
		No. of Rows=4
		No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The size of Matrix is:
		No. of Rows=4
		No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 0
	The size of Matrix is:
		No. of Rows=8
		No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 1
	The size of Matrix is:
		No. of Rows=11
		No. of Columns=11
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[0, 1, 0, 1]
[0, 1, 0, 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]
   ];

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions 

-- Then we compute the solution with
CharP.MXLSolve(F);

-- And we achieve the following information on the screen.
----------------------------------------
The size of Matrix is:
		No. of Rows=4
		No. of Columns=9
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The size of Matrix is:
		No. of Rows=3
		No. of Columns=9
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 0
	The size of Matrix is:
		No. of Rows=14
		No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 4
	The size of Matrix is:
		No. of Rows=27
		No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 0
	The size of Matrix is:
		No. of Rows=12
		No. of Columns=14
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 0
	The size of Matrix is:
		No. of Rows=19
		No. of Columns=15
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	The No. of Mutants found = 0
	The size of Matrix is:
		No. of Rows=14
		No. of Columns=15
Appling Gaussian Elimination...
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
Gaussian Elimination Completed.
	The variables found till now, if any are:
	[x[1], x[2], x[3], x[4]]
	Please Check the uniqueness of solution.
	The Given system of polynomials does not
	seem to have a unique solution.

See also

CharP.XLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.IMNLASolve

CharP.MNLASolve

CharP.NLASolve

CharP.IMXLSolve