#4. Let I = < f >, J = < g > be principal ideals in C[x_1...x_n].
a) Figure out what I intersect J is, given that it's supposed to give the union of the vanishing sets.
b) Follow the computation of (tI + (1-t)J) intersect C[x_1..x_n], explicitly, to see that you get the same answer as you did in part (a).
#5. Get Macaulay 2 running on some computer, to the extent that you can do the following computation. I've put [editorial comments] in the below to explain what's going on.
----------------------------------------------------------------------------
Macaulay2, version 1.4
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : R = QQ[m_(1,1)..m_(2,3)];
[R is the ring of polynomials with rational coefficients in six variables m_(i,j)]
i2 : M = transpose genericMatrix(R,m_(1,1),3,2)
[M is the matrix with the matrix entries m_(i,j) ]
i3 : I = ideal {det M_{0,1}, det M_{1,2}}
[M_{list} is the submatrix using only those columns, numbered 0,...,width-1 as usual in computer science. Here we're asking that columns 0 and 1, and columns 1 and 2, be linearly dependent.]
i4 : cs = decompose I
[We did this example in class a couple of weeks ago; this ideal I is not a prime ideal, but the intersection of two prime ideals. In one of them column 1 vanishes, in the other columns 0 and 2 are linearly dependent. "decompose" finds the two.]
i5 : intersect cs
i5 : intersect cs
[This just recovers I -- it is indeed the intersection of the two.]
