weightVectorsRealizingGB -- The main function for detecting Gröbner bases

Synopsis

Usage:

S = weightVectorsRealizingGB G
Inputs:
- G, 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 Gröbner basis

Description

We give three examples of ideal generators that are Gröbner bases for the indicated term orders below:

i1 : R1 = QQ[x,y, MonomialOrder=>Lex];

i2 : G1 = {y^2-x, x^2-1};

i3 : weightVectorsRealizingGB G1

o3 = {{2, 2}}

o3 : List

i4 : R2 = QQ[x,y,z, MonomialOrder=>Lex]

o4 = R2

o4 : PolynomialRing

i5 : G2 = {x^3-y, x^5-z}

       3       5
o5 = {x  - y, x  - z}

o5 : List

i6 : weightVectorsRealizingGB G2

o6 = {{2, 7, 5}, {2, 3, 11}, {1, 4, 6}}

o6 : List

i7 : R3 = QQ[x,y,z,w, MonomialOrder=>Lex]

o7 = R3

o7 : PolynomialRing

i8 : G3 = {x^2-z, x*y-z^2, x*z-y, w-z^2, y^2-z^3}

       2             2              2       2    3
o8 = {x  - z, x*y - z , x*z - y, - z  + w, y  - z }

o8 : List

i9 : weightVectorsRealizingGB G3

o9 = {{3, 8, 4, 4}, {2, 5, 2, 5}, {3, 4, 2, 5}, {1, 6, 3, 7}, {1, 8, 5, 11},
     ------------------------------------------------------------------------
     {4, 3, 3, 7}, {4, 3, 5, 11}, {1, 7, 5, 11}}

o9 : List

Here are two examples of generating sets which are not Gröbner bases:

i10 : R4 = QQ[x,y, MonomialOrder=>Lex];

i11 : G4 = {x^2+y^2-1, 2*x*y-1};

i12 : weightVectorsRealizingGB G4

o12 = {}

o12 : List

i13 : R5 = QQ[x,y,z, MonomialOrder=>Lex];

i14 : G5 = {x*y^2-x*z+y, x*y-z^2, x-y*z^4};

i15 : weightVectorsRealizingGB G5

o15 = {}

o15 : List

Ways to use weightVectorsRealizingGB :

weightVectorsRealizingGB(List)
weightVectorsRealizingGB(Matrix)

For the programmer

The object weightVectorsRealizingGB is a method function with options.