ApCoCoA-1:CharP.LASolve
CharP.LASolve
Computes the unique F_2-rational zero of a given polynomial system over F_2.
Syntax
CharP.LASolve(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 LA-Algorithm to find the unique zero. The LA-Algorithm generates a sequence of linear systems to solve the given system. The LA-Algorithm can find the unique zero only. If the given polynomial system has more than one zero's 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. To solve linear system naive Gaußian elimination is used.
@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.LASolve(F);
[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.NLASolve(F,Sparse);
x[4] = NA
Please Check the uniqueness of solution.
The Given system of polynomials does not
seem to have a unique solution or it has
no solution over the finite field F2.
See also
Introduction to Groebner Basis in CoCoA
