We now study the case of tensors.
Since the maximum rank of tensors is 5, the
maximum rank of
tensors is at least 5 and
at most 8.
[C-B-L-C2009] (table 1) states (as know previously) that the
typical rank is 5.
We choose ,
, and
randomly in
. Since these three tensors are in
, any tensor
on the line connecting any two of them is in
, and a tensor
in the plane is generically in
, which we is large
enough to capture general
tensors.
We first produce a visualization of on
, and obtain
[data file 3x3x4pr5random.dat]
Next, using the same set of random ,
, and
, we produce a visualization of
, and obtain
[data file 3x3x4pr6random.dat]
Initially the upper-right corner was generally lighter, so we added
more points on , which explains the greater
density in this region.
Based on these visualizations, we would conjecture that the maximum
rank of tensors is probably 5, and at most 6.
As noted above, [C-B-L-C2009] says it is known (proven) to be 5.