-- -*- M2-comint -*- {* hash: -1038181858 *}

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

i5 : leadTerm(1,I)

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

             1       7
o5 : Matrix R  <--- R

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

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

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

o15 = R

o15 : PolynomialRing

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

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 = | xs-yc xc2-x+ysc y2 xy x2 |

              1       5
o21 : Matrix B  <--- B

i22 : leadTerm oo

o22 = | xs xc2 y2 xy x2 |

              1       5
o22 : Matrix B  <--- B

i23 :