# Difference between revisions of "ApCoCoA-1: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]]
-------------------------------
```