Difference between revisions of "ApCoCoA-1:CharP.MBBasisF2"

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(New page: <command> <title>CharP.MBBasis</title> <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description> <syntax> CharP.MBBasisF2(F:LIST):LIST </...)
 
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 +
{{Version|1}}
 
<command>
 
<command>
 
     <title>CharP.MBBasis</title>
 
     <title>CharP.MBBasis</title>
−
     <short_description>Computing a Border Basis of a given ideal over <tt>F_2</tt>. </short_description>
+
     <short_description>Computes a Border Basis of a given ideal over <tt>F_2</tt>. </short_description>
 
<syntax>
 
<syntax>
 
CharP.MBBasisF2(F:LIST):LIST
 
CharP.MBBasisF2(F:LIST):LIST
 +
CharP.MBBasisF2(F:LIST, NSol: INT):LIST
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
 
<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.
+
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.
−
 
 
 
<itemize>
 
<itemize>
 
<item>@param <em>F:</em> List of polynomials.</item>
 
<item>@param <em>F:</em> List of polynomials.</item>
−
<item>@return A Border Basis of zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item>
+
<item>@param <em>NSol:</em> Number of <tt>F_2</tt> rational solutions. </item>
 +
<item>@return A Border Basis of the zero-dimensional radical ideal generated by the polynomials in F and the field polynomials. </item>
 
</itemize>
 
</itemize>
  
Line 25: Line 26:
 
     ];
 
     ];
  
−
-- Then we compute the solution with
+
-- Then we compute a Border Basis with
−
CharP.MXLSolve(F);
+
CharP.MBBasisF2(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
−
The size of Matrix is:
+
    No. of Columns=11
−
No. of Rows=4
+
    The size of Matrix is:
−
No. of Columns=11
+
    No. of Rows=8
−
Appling Gaussian Elimination...
+
    No. of Columns=11
−
-- CoCoAServer: computing Cpu Time = 0
+
    No. of mutants found =1
−
-------------------------------
+
    The size of Matrix is:
−
Gaussian Elimination Completed.
+
    No. of Rows=11
−
The size of Matrix is:
+
    No. of Columns=11
−
No. of Rows=4
+
    No. of mutants found =2
−
No. of Columns=11
+
    The size of Matrix is:
−
Appling Gaussian Elimination...
+
    No. of Rows=16
−
-- CoCoAServer: computing Cpu Time = 0
+
    No. of Columns=11
−
-------------------------------
+
    No. of mutants found =0
−
Gaussian Elimination Completed.
+
    The size of Matrix is:
−
The variables found till now, if any are:
+
    No. of Rows=31
−
[x[1], x[2], x[3], x[4]]
+
    No. of Columns=15
−
The No. of Mutants found = 0
+
    No. of mutants found =0
−
The size of Matrix is:
+
 
−
No. of Rows=8
+
[x[4] + 1, x[3], x[2] + 1, x[1]]
−
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>
 
</example>
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     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  
 
-- 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.MBBasisF2(F,NSol);
  
−
-- Then we compute the solution with
+
    The size of Matrix is:
−
CharP.MXLSolve(F);
+
    No. of Rows=4
−
 
+
    No. of Columns=9
−
-- And we achieve the following information on the screen.
+
    The size of Matrix is:
−
----------------------------------------
+
    No. of Rows=14
−
The size of Matrix is:
+
    No. of Columns=14
−
No. of Rows=4
+
    The size of Matrix is:
−
No. of Columns=9
+
    No. of Rows=16
−
Appling Gaussian Elimination...
+
    No. of Columns=15
−
-- CoCoAServer: computing Cpu Time = 0
+
[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]]
−
-------------------------------
 
−
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.
 
  
 
</example>
 
</example>
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     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
−
       <see>CharP.XLSolve</see>
+
       <see>ApCoCoA-1:CharP.MXLSolve|CharP.MXLSolve</see>
−
     <see>Introduction to CoCoAServer</see>
+
     <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
−
     <see>Introduction to Groebner Basis in CoCoA</see>
+
     <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
−
     <see>CharP.IMNLASolve</see>
+
     <see>ApCoCoA-1:CharP.IMNLASolve|CharP.IMNLASolve</see>
−
    <see>CharP.MNLASolve</see>
+
     <see>ApCoCoA-1:CharP.IMBBasisF2|CharP.IMBBasisF2</see>   
−
     <see>CharP.NLASolve</see>
 
−
    <see>CharP.IMXLSolve</see>   
 
 
   </seealso>
 
   </seealso>
  
Line 176: Line 97:
 
     </types>
 
     </types>
  
−
     <key>charP.mxlsolve</key>
+
     <key>charP.mbbasisf2</key>
−
     <key>mxlsolve</key>
+
     <key>mbbasisf2</key>
 
     <key>finite field</key>
 
     <key>finite field</key>
−
     <wiki-category>Package_charP</wiki-category>
+
     <wiki-category>ApCoCoA-1:Package_charP</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 09:56, 7 October 2020

This article is about a function from ApCoCoA-1.

CharP.MBBasis

Computes a Border Basis of a given ideal over F_2.

Syntax

CharP.MBBasisF2(F:LIST):LIST
CharP.MBBasisF2(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 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.MBBasisF2(F);

    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=11
    The size of Matrix is:
    	No. of Rows=8
    	No. of Columns=11
    No. of mutants found =1
    The size of Matrix is:
    	No. of Rows=11
    	No. of Columns=11
    No. of mutants found =2
    The size of Matrix is:
    	No. of Rows=16
    	No. of Columns=11
    No. of mutants found =0
    The size of Matrix is:
    	No. of Rows=31
    	No. of Columns=15
    No. of mutants found =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.MBBasisF2(F,NSol);

    The size of Matrix is:
    	No. of Rows=4
    	No. of Columns=9
    The size of Matrix is:
    	No. of Rows=14
    	No. of Columns=14
    The size of Matrix is:
    	No. of Rows=16
    	No. of Columns=15
[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.IMBBasisF2





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