# ApCoCoA-1:BB.BorderDivAlg

## BB.BorderDivAlg

Apply the border division algorithm.

### Syntax

```BB.BorderDivAlg(F:POLY,OO:LIST of POLY,Prebasis:LIST of POLY):RECORD
BB.BorderDivAlg(F:POLY,OO:LIST of POLY,Prebasis:LIST of LIST of POLY):RECORD```

### 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.

Applies the Border Division Algorithm w.r.t. the order ideal OO and the border prebasis

Prebasis to the polynomial F and returns a record with fields Quotients

and Remainder sucht that Remainder is the normal OO-remainder. Please note that you have to start the ApCoCoAServer in order to use this function.

As it is not immediately clear which term in the support of a given prebasis polynomial of

Prebasis is contained in the border of OO (remember that a term ordering is

used automatically), the prebasis needs to be parsed internally and a more detailed prebasis representation is computed. The internal expansion will be skipped if you already pass a more detailed prebasis description to the function which is possible by using the second function call (see example below).

• @param F The Border Division Algorithm will be applied to this polynomial.

• @param OO A list of terms representing an order ideal.

• @param Prebasis A list of polynomials representing a OO-border prebasis. Please see examples below for a detailed explanation of the format of this parameter.

• @return The result of the Border Divison Algorithm will be stored in a record containing two fields Quotients and Remainder, both of type POLY.

#### Example

```Use QQ[x,y];
OO := [1, x, y];
Prebasis := [ x^2 + x + 1, xy + y, y^2 + x + 1 ];
F := x^3y^2 - xy^2 + x^2 + 2;
BB.BorderDivAlg(F, OO, Prebasis);

-------------------------------
Record[Quotients = [xy^2 - y^2 + 1, -y, 2], Remainder = -3x - 1]
-------------------------------
```

#### Example

```-- The paramter Prebasis is internally expanded to
-- [ [ x^2 + x + 1, x^2 ], [ xy + y, xy ], [ y^2 + x + 1, y^2] ].
-- Thus, the following call of BB.BorderDivAlg is
-- equivalent to the one above
DetailedPrebasis := [ [ x^2 + x + 1, x^2 ], [ xy + y, xy ], [ y^2 + x + 1, y^2] ];
BB.BorderDivAlg(F, OO, DetailedPrebasis);

-------------------------------
Record[Quotients = [xy^2 - y^2 + 1, -y, 2], Remainder = -3x - 1]
-------------------------------
```