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# weightVectorsRealizingSAGBI -- The main function for detecting SAGBI bases

## Synopsis

• Usage:
S = weightVectorsRealizingSAGBI Q
• Inputs:
• Q, a list, of polynomials in the same ring
• Optional inputs:
• Verbose (missing documentation) => ..., default value false,
• Outputs:
• S, a list, (possibly empty) containing weight vectors for which Q forms a SAGBI basis

## Description

Here are two examples where we find one or more weight vectors for which the polynomials form a SAGBI basis.

 i1 : S1 = QQ[x,y,z, MonomialOrder=>Lex]; i2 : Q1 = {z, z*x, z*y, z*x*(x^2+y^2),z*y*(x^2+y^2)}; i3 : weightVectorsRealizingSAGBI Q1 o3 = {{2, 1, 1}, {1, 2, 1}} o3 : List i4 : S2 = QQ[x, MonomialOrder=>Lex]; i5 : Q2 = {x^4+x^3, x^2+x, x^3 + x^2}; i6 : weightVectorsRealizingSAGBI Q2 o6 = {{1}} o6 : List

Here is an example where the algebra generators are not a SAGBI basis for any term order.

 i7 : S3 = QQ[x,y, MonomialOrder=>Lex]; i8 : Q3 = {x+y, x*y, x*y^2}; i9 : weightVectorsRealizingSAGBI Q3 o9 = {} o9 : List

## Ways to use weightVectorsRealizingSAGBI :

• weightVectorsRealizingSAGBI(List)
• weightVectorsRealizingSAGBI(Matrix)

## For the programmer

The object weightVectorsRealizingSAGBI is .