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# 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 .