# ApCoCoA-1:CharP.IMNLASolve

## CharP.GBasisF2

Computing the unique `F_2-`rational zero of a given polynomial system over `F_2`.

### Syntax

CharP.IMNLASolve(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 Improved Mutant NLA`-`Algorithm to find the unique zero. The Improved Mutant NLA`-`Algorithm generates a sequence of linear systems to solve the given system. The Improved Mutant NLA`-`Algorithm can find the unique zero only. 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. In fact Improved Mutant NLA`-`Algorithm is the NLA`-`Algorithm with improved mutant strategy. It uses linalg.EF for gaussian elimination.

@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.IMNLASolve(F); -- And we achieve the following information on the screen together with the solution at the end. ---------------------------------------- Finding Variable: x[4] The size of Matrix is: No. of Rows=4 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=8 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 1 The total No. of Mutants found are = 1 The No. of Mutants of Minimum degree (Mutants used) are = 1 The size of Matrix is: No. of Rows=11 No. of Columns=11 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=11 No. of Columns=10 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=11 No. of Columns=10 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[4] = 1 Finding Variable: x[3] The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=4 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=7 No. of Columns=5 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=7 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=7 No. of Columns=7 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[3] = 0 Finding Variable: x[2] The size of Matrix is: No. of Rows=3 No. of Columns=4 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=3 No. of Columns=4 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=4 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=4 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. x[2] = 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.IMNLASolve(F); -- And we achieve the following information on the screen. ---------------------------------------- Finding Variable: x[4] The size of Matrix is: No. of Rows=4 No. of Columns=9 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=3 No. of Columns=9 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=9 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=9 No. of Columns=4 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=14 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=14 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 2 The total No. of Mutants found are = 2 The No. of Mutants of Minimum degree (Mutants used) are = 2 The size of Matrix is: No. of Rows=9 No. of Columns=10 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=7 No. of Columns=10 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=10 No. of Columns=8 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. No. of New Mutants found = 0 The size of Matrix is: No. of Rows=22 No. of Columns=14 Appling Gaussian Elimination for finding Mutants... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Compeleted. The size of Matrix is: No. of Rows=14 No. of Columns=13 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. The size of Matrix is: No. of Rows=14 No. of Columns=13 Appling Gaussian Elimination to check solution coordinate... -- CoCoAServer: computing Cpu Time = 0 ------------------------------- Gaussian Elimination Completed. 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