ApCoCoA-1:CharP.MXLSolve
From ApCoCoAWiki
Revision as of 08:43, 7 December 2010 by 132.231.10.53 (talk)
CharP.GBasisF2
Computing the unique F_2-rational zero of a given polynomial system over F_2.
Syntax
CharP.MXLSolve(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.
This function computes the unique zero in F_2^n of a polynomial system over F_2 . It uses Mutant XL-Algorithm to find the unique zero. The Mutant XL-Algorithm is impelemented only to find a unique solution. If the given polynomial system has more than one zeros in F_2^n then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound.
@param F: List of polynomials of given system.
@return The unique solution of the given system in F_2^n.
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
Introduction to Groebner Basis in CoCoA
