# ApCoCoA:BB.BBasisForMP

## 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]] -------------------------------