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

From ApCoCoAWiki
(Example section update.)
(Description update.)
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<command>
 
<command>
    <title>BB.BorderDivAlg</title>
+
  <title>BB.BorderDivAlg</title>
    <short_description>Apply the border division algorithm.</short_description>
+
  <short_description>Apply the border division algorithm.</short_description>
<syntax>
+
  <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 POLY):RECORD
+
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
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  <description>
</syntax>
 
    <description>
 
 
{{ApCoCoAServer}}
 
{{ApCoCoAServer}}
  
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and <tt>Remainder</tt> sucht that <tt>Remainder</tt> is the normal <tt>OO</tt>-remainder. Please
 
and <tt>Remainder</tt> sucht that <tt>Remainder</tt> is the normal <tt>OO</tt>-remainder. Please
 
note that you have to start the ApCoCoAServer in order to use this function.
 
note that you have to start the ApCoCoAServer in order to use this function.
 
+
<par/>
 
As it is not immediately clear which term in the support of a given prebasis polynomial of
 
As it is not immediately clear which term in the support of a given prebasis polynomial of
 
<tt>Prebasis</tt> is contained in the border of <tt>OO</tt> (remember that a term ordering is
 
<tt>Prebasis</tt> is contained in the border of <tt>OO</tt> (remember that a term ordering is
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-------------------------------
 
-------------------------------
 
</example>
 
</example>
    </description>
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  </description>
<types>
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  <types>
  <type>poly</type>
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    <type>poly</type>
  <type>polynomial</type>
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    <type>polynomial</type>
  <type>list</type>
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    <type>list</type>
  <type>apcocoaserver</type>
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    <type>apcocoaserver</type>
</types>
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  </types>
    <see>DivAlg</see>
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  <see>DivAlg</see>
    <key>BorderDivAlg</key>
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  <key>BorderDivAlg</key>
    <key>BB.BorderDivAlg</key>
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  <key>BB.BorderDivAlg</key>
    <key>borderbasis.BorderDivAlg</key>
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  <key>borderbasis.BorderDivAlg</key>
    <wiki-category>Package_borderbasis</wiki-category>
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  <wiki-category>Package_borderbasis</wiki-category>
 
</command>
 
</command>

Revision as of 10:45, 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