Chapter 9: Tensors

This chapter was presented by Paul Goerlach and Madeline Brandt. The code is written by Tim Seynnaeve in Macaulay2.

Exercise 3: Consider the \(3 \times 3 \times 2 \times 2\) tensor defined by the multilinear form \(x_1y_1z_1w_1+x_2y_2z_2w_2\). Determine all quadruples of singular vectors of \(T\).


--We compute the gradient of T
grad = jacobian ideal T

A point \(([x_1:x_2:x_3],[y_1:y_2:y_3],[z_1:z_2],[w_1:w_2])\) in \(\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^1\) is a singular tuples if it is fixed by the gradient map. To check if 2 points are equal in \(\mathbb{P}^{n-1}\), we view them as rows of a \(2 \times n\) matrix and ask for the rank to be 1.

M=(vars R)||(transpose grad)

The ideal I cuts out a subvariety of \(\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^1\) consisting of all singular tuples. We try to decompose it into its irreducible components:

decompose I

This decomposition contains several obsolete components. For example the ideal (x1,x2,x3): there are no points in \(\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^1\) with \(x_1=x_2=x_3=0\). This is because the ideal \((x_1,x_2,x_3)\) contains the “irrelevant ideal” of \(\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1 \times \mathbb{P}^1\). We can get rid of these obsolete components by saturating with respect to the irrelevant ideal:


Now we can analyze the components of our variety of singular tuples:

D=decompose I

We can look at their (co)dimensions and degrees:

apply(D,i->(codim i,degree i))

So there are 19 2-dimensional planes of singular tuples, plus 18 additional singular tuples, 10 of which are real and 8 of which are complex