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

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