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BB.BorderDivAlg

Applies 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.

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).

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]
-------------------------------


See Also