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

(Updated description. (Skaspar)) |
m (insert version info) |
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+ | {{Version|1}} | ||

<command> | <command> | ||

<title>BB.BBasisForMP</title> | <title>BB.BBasisForMP</title> | ||

− | <short_description>Computes the border basis of | + | <short_description>Computes the border basis of a zero-dimensional ideal generated by marked polynomials.</short_description> |

<syntax> | <syntax> | ||

Line 7: | Line 8: | ||

</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. | |

+ | <par/> | ||

+ | 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> | ||

Line 13: | Line 16: | ||

</itemize> | </itemize> | ||

<example> | <example> | ||

− | Use | + | 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> | ||

+ | <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]] | ||

------------------------------- | ------------------------------- | ||

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

Line 23: | Line 60: | ||

<type>ideal</type> | <type>ideal</type> | ||

</types> | </types> | ||

− | <see>BB.BBasis</see> | + | <see>ApCoCoA-1:BB.BBasis|BB.BBasis</see> |

− | <see>BB.BBasisForOI</see> | + | <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> | + | <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]] -------------------------------