# Difference between revisions of "ApCoCoA-1:CharP.IMBBasisF2"

(New page: <command> <title>CharP.IMBBasis</title> <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description> <syntax> CharP.IMBBasisF2(F:LIST):LIST ...) |
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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

<par/> | <par/> | ||

− | Let <tt>f_1</tt>, ... , <tt>f_m</tt> 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 <tt>f_1</tt>, ... , <tt>f_m</tt> and the field polynomials. Furthermore, it uses 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 <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions. | + | Let <tt>f_1</tt>, ... , <tt>f_m</tt> 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 <tt>f_1</tt>, ... , <tt>f_m</tt> 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 <tt>F_2</tt> rational solutions. The first version is safe to use if you do not know the exact number of <tt>F_2</tt> rational solutions. |

<itemize> | <itemize> | ||

<item>@param <em>F:</em> List of polynomials.</item> | <item>@param <em>F:</em> List of polynomials.</item> | ||

Line 26: | Line 26: | ||

-- Then we compute a Border Basis with | -- Then we compute a Border Basis with | ||

− | CharP. | + | CharP.IMBBasisF2(F); |

The size of Matrix is: | The size of Matrix is: | ||

Line 32: | Line 32: | ||

No. of Columns=11 | No. of Columns=11 | ||

The size of Matrix is: | The size of Matrix is: | ||

− | No. of Rows= | + | No. of Rows=4 |

No. of Columns=11 | No. of Columns=11 | ||

− | No. of | + | Total No. of Mutants are = 0 |

The size of Matrix is: | The size of Matrix is: | ||

− | No. of Rows= | + | No. of Rows=12 |

− | No. of Columns= | + | No. of Columns=15 |

− | No. of | + | 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: | The size of Matrix is: | ||

No. of Rows=16 | No. of Rows=16 | ||

− | No. of Columns= | + | No. of Columns=15 |

− | No. of | + | 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: | The size of Matrix is: | ||

− | No. of Rows= | + | No. of Rows=18 |

No. of Columns=15 | No. of Columns=15 | ||

− | No. of | + | Total No. of Mutants are = 0 |

[x[4] + 1, x[3], x[2] + 1, x[1]] | [x[4] + 1, x[3], x[2] + 1, x[1]] | ||

− | |||

</example> | </example> | ||

Line 66: | Line 82: | ||

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions | -- 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 | -- Compute the solution with | ||

− | CharP. | + | CharP.IMBBasisF2(F,NSol); |

The size of Matrix is: | The size of Matrix is: | ||

No. of Rows=4 | No. of Rows=4 | ||

No. of Columns=9 | 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: | The size of Matrix is: | ||

No. of Rows=14 | No. of Rows=14 | ||

No. of Columns=14 | No. of Columns=14 | ||

The size of Matrix is: | The size of Matrix is: | ||

− | No. of Rows= | + | No. of Rows=11 |

− | No. of Columns= | + | 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]] | [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]] | ||

Line 88: | Line 108: | ||

<see>Introduction to Groebner Basis in CoCoA</see> | <see>Introduction to Groebner Basis in CoCoA</see> | ||

<see>CharP.IMNLASolve</see> | <see>CharP.IMNLASolve</see> | ||

− | <see>CharP. | + | <see>CharP.MBBasisF2</see> |

</seealso> | </seealso> | ||

## Revision as of 13:59, 28 April 2011

## CharP.IMBBasis

Computing 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