simple Groebner basis computations over various rings

For a definition of Groebner basis, see what is a Groebner basis?.

A first Groebner basis

Weight vectors and non-strict monomial orders

The monomial order may be given by a weight vector (see monomial orderings),

i1 : R = QQ[a..f,MonomialOrder=>Weights=>{1,1,1,1,0,0}]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(a*b*c-d*e*f,a*c*e-b*d*f,a*d*f-b*c*e)

o2 = ideal (a*b*c - d*e*f, a*c*e - b*d*f, - b*c*e + a*d*f)

o2 : Ideal of R

i3 : gens gb I

o3 = | bce-adf ace-bdf abc-def b2df-de2f a2df-de2f c2de3f-d3ef3 abd2f2-cde3f
     ------------------------------------------------------------------------
     |

             1      7
o3 : Matrix R  <-- R

i4 : leadTerm I

o4 = | bce ace abc b2df a2df c2de3f abd2f2 |

             1      7
o4 : Matrix R  <-- R

The monomial order used here is: first consider the dot product of the weight vector and the exponent vectors. The larger determines the lead term. If these dot products are the same, then ties are broken with graded reverse lexicographic order. Sometimes, one wants the lead term ideal with respect to the weight vector itself. This ideal, often not a monomial ideal, may be obtained by selecting only the first part of the monomial order while finding the lead term matrix:

i5 : leadTerm(1,I)

o5 = | bce-adf ace-bdf abc b2df a2df c2de3f-d3ef3 abd2f2 |

             1      7
o5 : Matrix R  <-- R

Groebner basis over the integers

A strong Groebner basis of an ideal $I \subset{} ZZ[x_1,...,x_n]$ is a set $G$ of elements of $I$ such that, if $cm$ is any lead term of $I$ ($c$ a coefficient, and $m$ a monomial), then there is an element of $G$ whose lead term divides $cm$.

Macaulay2 computes such strong Groebner bases over ZZ.

i6 : R = ZZ[x,y]

o6 = R

o6 : PolynomialRing

i7 : F = y^2-(x^3+3*x+5)

        3    2
o7 = - x  + y  - 3x - 5

o7 : R

i8 : I = ideal(F, diff(x,F), diff(y,F))

               3    2               2
o8 = ideal (- x  + y  - 3x - 5, - 3x  - 3, 2y)

o8 : Ideal of R

i9 : gens gb I

o9 = | 174 2y 6x-72 y2+87 3x2+3 x2y+y x3+3x-82 |

             1      7
o9 : Matrix R  <-- R

i10 : leadTerm I

o10 = | 174 2y 6x y2 3x2 x2y x3 |

              1      7
o10 : Matrix R  <-- R

i11 : factor 174

o11 = 2*3*29

o11 : Expression of class Product

This elliptic curve is smooth for all primes $p \neq 2, 3, 29$.

Groebner basis over a quotient ring

The lead terms of the Groebner basis generate all possible lead terms modulo the lead terms of the quotient ideal.

i12 : R = QQ[a..d]/(a^2+b^2+c^2+d^2-1)

o12 = R

o12 : QuotientRing

i13 : I = ideal(a*b*c*d)

o13 = ideal(a*b*c*d)

o13 : Ideal of R

i14 : gens gb I

o14 = | abcd b3cd+bc3d+bcd3-bcd |

              1      2
o14 : Matrix R  <-- R

A Groebner basis over a skew-commutative ring

The lead terms of the Groebner basis generate all possible lead terms, as in the commutative case. Essentially the same algorithm is applicable in this case.

i15 : R = QQ[a..d,SkewCommutative=>true]

o15 = R

o15 : PolynomialRing, 4 skew commutative variable(s)

i16 : I = ideal(a*b-c*d)

o16 = ideal(a*b - c*d)

o16 : Ideal of R

i17 : gens gb I

o17 = | ab-cd bcd acd |

              1      3
o17 : Matrix R  <-- R

Groebner basis over polynomial rings over polynomial rings

For such rings, we may imagine a polynomial ring over a base field, or over the integers, possibly modulo some ideal. The resulting Groebner basis is this one. The monomial order is a product order, with the variables in the outermost polynomial ring the most expensive.

i18 : A = QQ[s,c]/(s^2+c^2-1)

o18 = A

o18 : QuotientRing

i19 : B = A[x,y,z]

o19 = B

o19 : PolynomialRing

i20 : I = ideal(c*x^2, s*y^2, c*y-s*x)

                2     2
o20 = ideal (c*x , s*y , - s*x + c*y)

o20 : Ideal of B

i21 : gens gb I

o21 = | sx-cy (c2-1)x+scy y2 xy x2 |

              1      5
o21 : Matrix B  <-- B

i22 : leadTerm oo

o22 = | sx c2x y2 xy x2 |

              1      5
o22 : Matrix B  <-- B