Difference between revisions of "ApCoCoA-1:SB.ReducedSagbi"
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(New page: <command> <title>SB.ReducedSagbi</title> <short_description>Computes the finite reduced SAGBI-basis of a subalgebra if existing.</short_description> <syntax> SB.ReducedSagbi(G:LIST ...) |
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| + | {{Version|1|[[Package sagbi/SB.SAGBI]]}} | ||
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If a finite SAGBI-basis of the subalgebra <tt>S</tt> generated by <tt>G</tt> is existing, this function computes the reduced SAGBI-basis of <tt>S</tt>. Then a list of polynomials is returned which form the reduced SAGBI-basis of <tt>S</tt>, i.e. these polynomials fulfill the conditions of a reduced SAGIB-basis. If no finite SAGBI-basis is existing the computation will be interrupted by an error message. | If a finite SAGBI-basis of the subalgebra <tt>S</tt> generated by <tt>G</tt> is existing, this function computes the reduced SAGBI-basis of <tt>S</tt>. Then a list of polynomials is returned which form the reduced SAGBI-basis of <tt>S</tt>, i.e. these polynomials fulfill the conditions of a reduced SAGIB-basis. If no finite SAGBI-basis is existing the computation will be interrupted by an error message. | ||
| + | <par/> | ||
| + | <em>Important:</em> This functions works only, if a finite SAGBI-basis of <tt>S</tt> is existing! | ||
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| − | <see>SB.Sagbi</see> | + | <see>ApCoCoA-1:SB.Sagbi|SB.Sagbi</see> |
| − | <see>SB.IsSagbi</see> | + | <see>ApCoCoA-1:SB.IsSagbi|SB.IsSagbi</see> |
| − | <see>SB.IsSagbiOf</see> | + | <see>ApCoCoA-1:SB.IsSagbiOf|SB.IsSagbiOf</see> |
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<type>sagbi</type> | <type>sagbi</type> | ||
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<key>sagbi.reducedsagbi</key> | <key>sagbi.reducedsagbi</key> | ||
| − | <wiki-category>Package_sagbi</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_sagbi</wiki-category> |
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Latest revision as of 18:05, 27 October 2020
| This article is about a function from ApCoCoA-1. If you are looking for the ApCoCoA-2 version of it, see Package sagbi/SB.SAGBI. |
SB.ReducedSagbi
Computes the finite reduced SAGBI-basis of a subalgebra if existing.
Syntax
SB.ReducedSagbi(G:LIST of POLY):LIST of POLY
Description
If a finite SAGBI-basis of the subalgebra S generated by G is existing, this function computes the reduced SAGBI-basis of S. Then a list of polynomials is returned which form the reduced SAGBI-basis of S, i.e. these polynomials fulfill the conditions of a reduced SAGIB-basis. If no finite SAGBI-basis is existing the computation will be interrupted by an error message.
Important: This functions works only, if a finite SAGBI-basis of S is existing!
@param G A list of polynomials which generates a subalgebra.
@return If no error will occur, a list of polynomials which form a reduced SAGBI-basis of the given subalgebra.
Example
Use R::=QQ[x,y]; G:=[x-y,x+y]; SB.Sagbi(G); SB.ReducedSagbi(G); ------------------------------------------------------- -- output: -- a SAGBI-basis of K[G] [ x - y, x + y, y] ------------------------------- -- the reduced SAGBI-basis of K[G] [ x, y] ------------------------------- -- Done. -------------------------------
Example
Use R::=QQ[x[1..6]];
G:=[-x[4] - x[6], -x[1], x[2]^2 + x[3]^2,
-4x[4]^2 - 5/2x[5]^2 + 2x[4]x[6] - 4x[6]^2,
-2x[2]x[3]x[4] + x[2]^2x[5] - x[3]^2x[5] + 2x[2]x[3]x[6],
-x[2]^2x[4] + x[3]^2x[4] - 2x[2]x[3]x[5] + x[2]^2x[6] - x[3]^2x[6]];
SB.Sagbi(G);
SB.ReducedSagbi(G);
-------------------------------------------------------
-- output:
-- a SAGBI-basis of K[G]
[
x[4] + x[6],
x[1],
x[2]^2 + x[3]^2,
x[4]^2 + 5/8x[5]^2 - 1/2x[4]x[6] + x[6]^2,
x[2]x[3]x[4] - 1/2x[2]^2x[5] + 1/2x[3]^2x[5] - x[2]x[3]x[6],
x[2]^2x[4] - x[3]^2x[4] + 2x[2]x[3]x[5] - x[2]^2x[6] + x[3]^2x[6],
x[5]^2 - 4x[4]x[6],
x[3]^2x[4] - x[2]x[3]x[5] + x[2]^2x[6]]
-------------------------------
-- the reduced SAGBI-basis of K[G]
[
x[4] + x[6],
x[1],
x[2]^2 + x[3]^2,
x[2]x[3]x[4] - 1/2x[2]^2x[5] + 1/2x[3]^2x[5] - x[2]x[3]x[6],
x[5]^2 - 4x[4]x[6],
x[3]^2x[4] - x[2]x[3]x[5] + x[2]^2x[6]]
-------------------------------
-- Done.
-------------------------------
Example
Use R::=QQ[x,y];
G:=[x+y, xy, xy^2];
SB.Sagbi(G);
SB.ReducedSagbi(G);
-------------------------------------------------------
-- output:
--- Computation was interrupted ---
NULL
-------------------------------
ERROR: SB.ReducedSagbi: No finite SAGBI-Basis is existing!
CONTEXT: Error("SB.ReducedSagbi: No finite SAGBI-Basis is existing!")
-------------------------------
-- Done.
-------------------------------
