MA341D Homework assignment 5
Due on April 7, 2016
This homework is optional. If you choose to work on it, it may be used to make up for up
to one-tenth of the continuous assessment mark: each of the previous four homeworks is worth
25% of the overall continuous assessment mark; this homework may add up to 10% to that.
However, if you choose to not hand it in, you will not be penalised in any way. You may use
computer algebra systems, in which case you are required to attach a printout of the programs
you executed.
Let us consider Chx, yi equipped with the glex order with x < y. We take some a, b, c, d ∈ C
which are not simultaneously equal to zero, and consider the element
f = ay 2 + byx + cxy + dx2 ∈ Chx, yi.
We denote by I the ideal generated by that element, I = (f ). In this problem sheet, our goal is
to learn something about how the dimension of the n-th homogeneous component of Chx, yi/I
depends on a, b, c, d.
1. (25 marks) Compute the dimension of the n-th homogeneous component of Chx, yi/I for
(i) a = b = c = 0;
(ii) a = b = 0, c 6= 0;
(iii) a = 0, b 6= 0.
Important: in the remaining questions, we assume that a = 1 (which we may assume
without loss of generality in the case a 6= 0).
2. (10 marks) Denote by I(k) the subspace of I that consists of homogeneous elements of
degree exactly k. Show that dim I(1) = 0, dim I(2) = 1, dim I(3) ≤ 4, and more generally
dim I(k) ≤ (k − 1)2k−2 .
Write down some four noncommutative polynomials that span I(3) .
In the following question, you may (but do not have to) use without proof that for a
rectangular matrix A, the rank of A is equal to k if and only if there exists a k × k-minor
of A that is different from zero, and all (k + 1) × (k + 1)-minors of A are equal to zero.
3. (25 marks) Write down explicitly polynomial equations for b, c, d which guarantee that
dim I(3) = 3 (and hence the dimension of the third homogeneous component of Chx, yi/I
is equal to 8 − 3 = 5).
4. (40 marks) Solve the equations that you wrote down in the previous question. For all
values of b, c, d you found as solutions, compute the dimension of the n-th homogeneous
component of Chx, yi/I.