ApCoCoA-1:SB.ReducedSagbi

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.
-------------------------------

SB.Sagbi

SB.IsSagbi

SB.IsSagbiOf