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# weightVectorsRealizingSAGBI -- The main function for detecting SAGBI bases

- 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 a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SagbiGbDetection.m2:215:0.