Difference between revisions of "ApCoCoA-1:BB.BorderDivAlg"

From ApCoCoAWiki
(Description update.)
(Description update.)
Line 5: Line 5:
 
BB.BorderDivAlg(F:POLY,OO:LIST of POLY,Prebasis:LIST of LIST of POLY):RECORD</syntax>
 
BB.BorderDivAlg(F:POLY,OO:LIST of POLY,Prebasis:LIST of LIST of POLY):RECORD</syntax>
 
   <description>
 
   <description>
{{ApCoCoAServer}}
+
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 
+
<par/>
 
Applies the Border Division Algorithm w.r.t. the order ideal <tt>OO</tt> and the border prebasis
 
Applies the Border Division Algorithm w.r.t. the order ideal <tt>OO</tt> and the border prebasis
 
<tt>Prebasis</tt> to the polynomial <tt>F</tt> and returns a record with fields <tt>Quotients</tt>
 
<tt>Prebasis</tt> to the polynomial <tt>F</tt> and returns a record with fields <tt>Quotients</tt>

Revision as of 11:05, 24 April 2009

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

DivAlg