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