Difference between revisions of "ApCoCoA-1:BB.BBasisForMP"

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{{Version|1}}
 
<command>
 
<command>
 
   <title>BB.BBasisForMP</title>
 
   <title>BB.BBasisForMP</title>
   <short_description>Computes the border basis of an ideal generated by marked polynomials.</short_description>
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   <short_description>Computes the border basis of a zero-dimensional ideal generated by marked polynomials.</short_description>
 
    
 
    
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
   <description>
 
   <description>
Computes the border basis of the ideal <tt>I = &lt;F&gt;</tt> with respect to the set of marked polynomials <tt>F</tt>. The input is a list of tuples <tt>[P, T]</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> must be a term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt>. The output is a list of tuples <tt>[P, T]</tt> denoting a border basis of <tt>I</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> is the term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt> and <tt>T</tt> is a border term. An error will indicate if the given term marking does not lead to a successful computation.
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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.
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<par/>
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The input is a list of tuples <tt>[P, T]</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> must be a term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt>. This function computes the border basis of the zero-dimensional ideal <tt>I</tt> generated by the polynomials <tt>P</tt> with respect to the given term marking. The output is a list of tuples <tt>[P, T]</tt> denoting a border basis of <tt>I</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> is the term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt> and <tt>T</tt> is a border term. An error will be raised if the given term marking does not lead to a successful computation.
 
<itemize>
 
<itemize>
 
   <item>@param <em>F</em> List of tuples <tt>[P, T]</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> must be a term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt>. The polynomials <tt>P</tt> must generate a zero-dimensional ideal.</item>
 
   <item>@param <em>F</em> List of tuples <tt>[P, T]</tt> where <tt>P</tt> is a polynomial and <tt>T</tt> must be a term of the support of <tt>P</tt> such that <tt>deg(P) = deg(T)</tt>. The polynomials <tt>P</tt> must generate a zero-dimensional ideal.</item>
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</itemize>
 
</itemize>
 
<example>
 
<example>
Use QQ[x,y];
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Use Q[x,y], DegLex;
BB.BBasisForMP([ [,], [,] ]);
 
  
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F := [
 +
[ x^2 + xy - 1/2y^2 - x - 1/2y, xy ],
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[ y^3 - y, y^3 ],
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[ xy^2 - xy, xy^2 ]
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];
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 +
BB.BBasisForMP(F);
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[[x^2 + xy - 1/2y^2 - x - 1/2y, xy],
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[y^3 - y, y^3],
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[xy^2 + x^2 - 1/2y^2 - x - 1/2y, xy^2],
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[x^3 - x, x^3],
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[x^2y - 1/2y^2 - 1/2y, x^2y]]
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-------------------------------
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</example>
 +
<example>
 +
Use Q[x,y,z], DegLex;
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 +
F := [
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[ x^2 + xy + y^2 - x - 1, x^2 ],
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[ xy + y^2 + z, xy ],
 +
[ -x^2 + yz + z + 1, x^2 ]
 +
];
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BB.BBasisForMP(F);
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[[x^2 - x - z - 1, x^2],
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[xy + z^2 + x + z + 1, xy],
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[yz - x, yz],
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[y^2 - z^2 - x - 1, y^2],
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[x^2z - xz - z^2 - z, x^2z],
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[xz^2 + xz - z^2 + 2x + y, xz^2],
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[xyz - x - z - 1, xyz],
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[z^3 + xz + z^2 + x + 2z + 1, z^3],
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[yz^2 - xz, yz^2]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
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     <type>ideal</type>
 
     <type>ideal</type>
 
   </types>
 
   </types>
   <see>BB.BBasis</see>
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   <see>ApCoCoA-1:BB.BBasis|BB.BBasis</see>
   <see>BB.BBasisForOI</see>
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   <see>ApCoCoA-1:BB.BBasisForOI|BB.BBasisForOI</see>
 
   <key>BBasisForMP</key>
 
   <key>BBasisForMP</key>
 
   <key>BB.BBasisForMP</key>
 
   <key>BB.BBasisForMP</key>
 
   <key>borderbasis.BBasisForMP</key>
 
   <key>borderbasis.BBasisForMP</key>
   <wiki-category>Package_borderbasis</wiki-category>
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   <wiki-category>ApCoCoA-1:Package_borderbasis</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:39, 7 October 2020

This article is about a function from ApCoCoA-1.

BB.BBasisForMP

Computes the border basis of a zero-dimensional ideal generated by marked polynomials.

Syntax

BB.BBasisForMP(F:LIST of LIST):LIST of 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.

The input is a list of tuples [P, T] where P is a polynomial and T must be a term of the support of P such that deg(P) = deg(T). This function computes the border basis of the zero-dimensional ideal I generated by the polynomials P with respect to the given term marking. The output is a list of tuples [P, T] denoting a border basis of I where P is a polynomial and T is the term of the support of P such that deg(P) = deg(T) and T is a border term. An error will be raised if the given term marking does not lead to a successful computation.

  • @param F List of tuples [P, T] where P is a polynomial and T must be a term of the support of P such that deg(P) = deg(T). The polynomials P must generate a zero-dimensional ideal.

  • @return A list of tuples [P, T] denoting a border basis of I where P is a polynomial and T is the term of the support of P such that deg(P) = deg(T) and T is a border term.

Example

Use Q[x,y], DegLex;

F := [
[ x^2 + xy - 1/2y^2 - x - 1/2y, xy ],
[ y^3 - y, y^3 ],
[ xy^2 - xy, xy^2 ]
];

BB.BBasisForMP(F);

[[x^2 + xy - 1/2y^2 - x - 1/2y, xy],
 [y^3 - y, y^3],
 [xy^2 + x^2 - 1/2y^2 - x - 1/2y, xy^2],
 [x^3 - x, x^3],
 [x^2y - 1/2y^2 - 1/2y, x^2y]]
-------------------------------

Example

Use Q[x,y,z], DegLex;

F := [
[ x^2 + xy + y^2 - x - 1, x^2 ],
[ xy + y^2 + z, xy ],
[ -x^2 + yz + z + 1, x^2 ]
];

BB.BBasisForMP(F);

[[x^2 - x - z - 1, x^2],
 [xy + z^2 + x + z + 1, xy],
 [yz - x, yz],
 [y^2 - z^2 - x - 1, y^2],
 [x^2z - xz - z^2 - z, x^2z],
 [xz^2 + xz - z^2 + 2x + y, xz^2],
 [xyz - x - z - 1, xyz],
 [z^3 + xz + z^2 + x + 2z + 1, z^3],
 [yz^2 - xz, yz^2]]
-------------------------------

BB.BBasis

BB.BBasisForOI