Just a short post: David Kazhdan, Tammy Ziegler and I have recently (and finally) posted our proof that the analytic rank of a cubic, the logarithm of its bias, and its Schmidt rank (how hard it is to write it as products of lower degree polynomials), are linearly related.
This came out of our discussion of Kronecker’s lemma. One version can be stated as follows:
Lemma (Kronecker) Consider A, B linear maps from a vectorspace X to another one Y over any infinite field. Assume that
B ker(A) is not a subspace of im A
then the generic linear combination of A and B has larger rank than A and B
I had used this lemma in a previous work to construct Lefschetz elements, without knowing at first it was due to Kronecker. More about that Lefschetz aspect later, when some exciting news will come online. Suffices to say, that Kronecker’s lemma gives a rather cool restriction on spaces of linear maps, and strongly restricts them when they are of low rank.
Corollary Consider a space of bilinear forms L over X times Y, such that every element is of rank at most r. Then there are subspaces X’ and Y’, each of codimension r in X resp. Y, so that L vanishes identically on X’ times Y’.
It isn’t much, really, but at first this corollary was quite counterintuitive to me, since after knowing that a space of matrices of rank r cannot be restricted to r rows or r columns (consider ) I did not believe anything of the sort could be true.
Anyway, for the application to tensor ranks, see our paper. Let me advertize also the upcoming Hadamard lectures, where I will explain how Kronecker’s lemma can be used to construct Lefschetz isomorphisms.
Kronecker, L. Algebraische Reduction der Scharen bilinearer Formen, Sitzungsber. Akad.Berlin, Jbuch. 22 (169) (1890) 1225-1237.