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I haven't downvoted, but that style of writing does my head in. The lack of a precise reference/citation to Harris's book does not help.
@Andy: insert the hallowed response of the Mexican bandits to Hedley Lamarr here...
Touche. I think Sierra Madre has been mentioned before on MMO.
Thank you all for your considerate remarks. Andy, my numerical investigations indicate that multilinear Segre immersions for which the TM nullspace is 1-dimensional (with respect to the immersing linear manifold) can be constructed as tri-linear and 5-, 6-, 10-linear examples; for these high-order varieties the bilinear analysis methods that commonly are taught at the undergraduate level of course fail completely — so perhaps I will post a future MOF question about the properties of these remarkable algebraic/geometric objects. And Yemon, it was no trouble to add the page-number references you requested.
Most of all, it genuinely was fun to help Theo win MOL's the first Gold Reversal Badge. Thank you all for working to make MOF a great resource, not only for mathematicians, but for us engineers and medical researchers too.
Here are my two cents:
1¢ Yay badge!
2¢ My understanding of the question as now edited is for information about generalizations to higher-order tensors of the trivial linear-algebraic fact that John Sidles calls "Second Hand Lion Theorem" and that I rephrased in my answer. Unfortunately, I don't have much to say about such generalizations, except that I agree with Harris's appraisal that they are much harder. It's not entirely clear to me what types of generalizations John is looking for, but that might be precisely the point of the question. Anyway, the question is by now long enough that it's a bit hard to scan. Depending on what John wants, I could imagine myself or someone else helping to write a new short version of this question, phrased purely in the language of mathematics, and closing the present question as "no longer relevant" with a link to the new one. I don't know what phrase will currently go on the question if it is closed, but I think it now does refer to some deep mathematics, so probably really fixing it the "correct" way will require some moderator magic.
Theojf, thank you for your well-considered summary remarks.
I have this evening numerically constructed several higher order examples of this class of multilinear varieties, and in the coming week or two, I very likely pose some sort of concrete question that suggested by their existence, and associated to their ruled algebraic structure, and related to their geometry as dynamical state-spaces.
Here is an example: for indices $\{s,r,m,n,o\}$ that range over $s \in \{1,\dots,3\}$, $r \in \{1,\dots,18\}$, and $m,n,o \in \{1,\dots,7\}$ we have
$$\psi_{(mno)} = \xi_{(1rm)}\,\xi_{(2rn)}\,\xi_{(3ro)}$$
as a variety of (numerically computed) dimension $342$ having a natural Segre immersion in a linear (Hilbert) space of dimension $7^3 =343$, so that the dimensional defect $343-342$ is unity (this unity dimensional defect being the defining characteristic of this class of algebraic objects).
Where do these unity-defect “almost-Hilbert” varieties come from? Do they belong to any known classification of geometric objects? While these higher-order geometric questions slowly clarify themselves, perhaps it is appropriate that the present low-order question simply rest fallow.
As a final remark, last night I discovered a brand-new monograph by Joseph Landsberg that encompasses more-or-less the answer sought, and so I have amended the beginning of the question to be a pointer to Landsberg's monograph.
Sometime in the next week or two I will post a concrete mathematical question — framed within the context that Landsberg's monograph supplies — asking for a classification of all multilinear varieties having unit-dimension defect with respect to their natural Segre embedding.
At that time I will request closure of the original question, to be supplanted by this classification question.
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