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

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(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 ...)
 
Line 9: Line 9:
 
<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>
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-- Then we compute a Border Basis with
 
-- Then we compute a Border Basis with
CharP.MBBasisF2(F);
+
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=8
+
     No. of Rows=4
 
     No. of Columns=11
 
     No. of Columns=11
     No. of mutants found =1
+
     Total No. of Mutants are = 0
 
     The size of Matrix is:
 
     The size of Matrix is:
     No. of Rows=11
+
     No. of Rows=12
     No. of Columns=11
+
     No. of Columns=15
     No. of mutants found =2
+
     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=11
+
     No. of Columns=15
     No. of mutants found =0
+
     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=31
+
     No. of Rows=18
 
     No. of Columns=15
 
     No. of Columns=15
     No. of mutants found =0
+
     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>
  
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-- 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.MBBasisF2(F,NSol);
+
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=16
+
     No. of Rows=11
     No. of Columns=15
+
     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]]
  
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     <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.IMBBasisF2</see>   
+
     <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

CharP.MXLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.IMNLASolve

CharP.MBBasisF2