Symmetric tensors: rank, Strassen's conjecture and e-computability

Counterexamples to Strassen’s direct sum conjecture

“…This problem remains open for d 4, and the author is not sure that the present approach can lead to a progress on the multidimensional version. The analogous statement but restricted to symmetric tensors is open already for d 3; see [13], [14], [45] for a review of the current state of art and new results on this problem. As said above, the border rank version of the direct sum conjecture is known to fail for d-tensors at least when d=3; see also a discussion in Chapter 11 of [25].…”

“…This approach has led to interesting generalizations and relaxations of Conjecture 1, to further sufficient conditions for tensors to satisfy it, and, therefore, to many new classes of tensors for which the conjecture holds. Let us mention the paper [14] on the so-called e-computable tensors, the work [13] devoted to symmetric tensors, the paper [45] dealing with the cactus rank and catalecticant bound, the work [27] proving Conjecture 1 for tensors whose ranks can be computed by a particular adaptation of the so-called substitution method, the paper [15] studying the spaces of feasible rank decompositions in context of the direct sum conjecture, the work [12] containing further reformulations and generalizations of Conjecture 1 in terms of secant varieties, the monograph [25] containing a detailed discussion of this conjecture and its consequences for algebraic geometry, and a recent survey paper [16] on the topic. Different rank-decomposition problems do also take an important place in linear algebra and combinatorics.…”

A note on the simultaneous Waring rank of monomials

“…In order to show the statement, we have to establish a lower bound on the cardinality of X. We adapt the proof of [10,Theorem 3.3]. Note that the simultaneous rank is at least the dimension of the algebra T /(I X : (L) + L) for any linear form L. Let J = ∩ j∈J F ⊥ j .…”

Sufficient conditions for Strassen’s additivity conjecture

“…. , ℓ r are general linear forms in n variables and (4,4), (5,4), (5,3) then rk(F ) = 6, 10, 15, 8, respectively (instead of 5, 9, 14, 7, respectively).…”

“…When all but one of the F i has the simple form x d i , or when the number of summands is k = 2 and each F 1 , F 2 is a binary form, Question 1 has a positive answer [7]. These cases and many more are treated in a more uniform manner in [5], which introduces the notion of "e-computability" and shows it provides a sufficient condition for Question 1 to have a positive answer. Very recently a positive answer to Question 2 for certain forms F i has been given in [9,Theorem 4.6,Theorem 5.1].…”