# ApCoCoA:BB.BBasisForMP

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<title>BB.BBasisForMP</title> <short_description>Computes the border basis of a zero-dimensional ideal generated by marked polynomials.</short_description>

<syntax> BB.BBasisForMP(F:LIST of LIST):LIST of LIST </syntax>

<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.
<par/>
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.
<itemize>

<item>@paramFList of tuples[P, T]wherePis a polynomial andTmust be a term of the support ofPsuch thatdeg(P) = deg(T). The polynomialsPmust generate a zero-dimensional ideal.</item> <item>@return A list of tuples[P, T]denoting a border basis ofIwherePis a polynomial andTis the term of the support ofPsuch thatdeg(P) = deg(T)andTis a border term.</item>

</itemize> <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> <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>

</description> <types> <type>borderbasis</type> <type>ideal</type> </types> <see>BB.BBasis</see> <see>BB.BBasisForOI</see> <key>BBasisForMP</key> <key>BB.BBasisForMP</key> <key>borderbasis.BBasisForMP</key> <wiki-category>Package_borderbasis</wiki-category>

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