I'm not all that familiar with commutative algebra, so I need help with the calculation of a Gröbner basis. Let $ k $ be a field, and consider $ R = k[x_{1},x_{2},x_{3},x_{4}] $. I need to find the reduced Gröbner basis for the intersection of the two ideals $ I = \langle x_{1} x_{3},x_{1} x_{4},x_{2} x_{3},x_{2} x_{4} \rangle $ and $ J = \langle x_{1} - x_{3},x_{2} - x_{4} \rangle $ of $ R $, with respect to the lexicographic monomial ordering $ x_{1} > x_{2} > x_{3} > x_{4} $. Thanks.
Calculation of Gröbner basis.
-
0@Andrew how do they compare with Singular when the polynomials are structure functions about sp-graphs (describing whether the system works or not)? – 2016-01-18
1 Answers
Here is my attempt at computing this using Macaulay2. Disclaimer: I am still quite new to this, so I can't guarantee correctness (although I didn't purposely do anything wrong ;-) ). In particular, it seems Macaulay2 does not use Lex ordering by default, hence the optional command in the definition of $R=k[a,b,c,d].$
It appears that we obtain $(b^2d-bd^2, b^2c-bcd, ad-bc, abc-bc^2, a^2c-ac^2)$ as the reduced Groebner basis for the intersection $I\cap J.$
i1 : R=QQ[a..d, MonomialOrder => Lex]
o1 = R
o1 : PolynomialRing
i2 : I=ideal(a*c,a*d,b*c,b*d)
o2 = ideal (a*c, a*d, b*c, b*d)
o2 : Ideal of R
i3 : J=ideal(a-c,b-d)
o3 = ideal (a - c, b - d)
o3 : Ideal of R
i4 : K=intersect(I,J)
2 2 2 2 2 2
o4 = ideal (a*d - b*c, b d - b*d , b c - b*c*d, a*b*c - b*c , a c - a*c )
o4 : Ideal of R
i5 : gens gb K
o5 = | b2d-bd2 b2c-bcd ad-bc abc-bc2 a2c-ac2 |
1 5
o5 : Matrix R <--- R
-
1Dear @HaskellCurry, haha, you are most welcome! Actually, I like to use Wolfram Alpha, but I never thought of using it to compute a Groebner basis, so I'm glad to know it can do that! I'm also glad that we got the same answer! Cheers – 2012-10-25