# Difference between revisions of "ApCoCoA-1:SB.ReducedSagbi"

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