We give an upper bound for the cactus rank of any multi-homogeneous polynomial.2010 Mathematics Subject Classification. 15A69, 14M12.

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Abstract. We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated in [13,11]. In particular, we (i) exhibit many non-obvious equations testing for (border) rigidity, (ii) compute degrees of varieties associated to rigidity, (iii) describe algebraic varieties associated to families of matrices that are expected to have super-linear rigidity, and (iv) prove results about the ideals and degrees of cones that are of interest in their own right.

Given tensors T and T of order k and k respectively, the tensor product T ⊗ T is a tensor of order k +k . It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called W -states, namely monomials of the form x d−1 y, and on products of such. In particular, we prove that the partially symmetric rank oftools from algebraic geometry, we improve upon these bounds and provide a number other insights on the rank of tensor products of symmetric tensors.1.1. Motivations. Tensor decomposition for structured tensors is a classical topic that has been studied in algebraic geometry at least since the nineteenth century and finds numerous applications in other fields, such as quantum physics and theoretical computer science. We present some of the applications in related fields.Entanglement. The Hilbert space of a composite quantum system is the tensor product of the Hilbert spaces of the constituent systems. The Hilbert space of the N -body system is obtained as the tensor product of N copies of the n-dimensional single particle Hilbert space H 1 . In the case of indistinguishable bosonic particles, the totally symmetric states under particle exchange are physically relevant, which amounts to restricting the attention to the subspace H s = S N H 1 ⊂ N H 1 of completely symmetric tensors. In case we have two different species of indistinguishable bosonic particles, the relevant Hilbert space is S N1 H 1 ⊗ S N2 H 2 . Tensor rank is a natural measure of the entanglement of the corresponding quantum state ([YCGD10], [BC12]) and strict submultiplicativity of partially symmetric rank reflects the unexpected fact that entanglement does not simply "add up" in the composite system formed by multiple bosonic systems, even if the statesThe results of this paper expand on this novel quantum effect. Communication Complexity. The log-rank of the communication matrix is a lower bound on the deterministic communication complexity (see [MS82]) and it is an open question whether this bound is tight up to polynomial factors ([LS88]). Recently, it has been shown that support tensor rank equals the non-deterministic multiparty quantum communication complexity in the quantum broadcast model ([BCZ17]). Here, the communicating parties obtain each an input and are asked to compute a Boolean function of the joint input using as little quantum communication as possible. The tensor encodes the Boolean function; the order of the tensor corresponds to the number of parties. Support tensor rank is upper bounded by tensor rank with equality in some cases: for instance, in the case of W -states or asymptotically in the equali...

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.The tensor border rank (border rank, for short) of T is R(T ) = min r : T = lim ε→0 T ε where, for every ε, R(T ε ) = r and the limit is taken in the Euclidean topology of V 1 ⊗ · · · ⊗ V k . Clearly R(T ) ≤ R(T ) and there are many examples where the inequality is strict.It is straightforward to verify that rank and border rank are submultiplicative under the tensor product: if T 1 and T 2 are tensors of order k 1 , k 2 respectively, then T 1 ⊗ T 2 is a tensor of order k 1 +k 2 satisfying R(T 1 ⊗T 2 ) ≤ R(T 1 )R(T 2 ) and R(T 1 ⊗T 2 ) ≤ R(T 1 )R(T 2 ). Recently, [CJZ18] answered a question posed in [Dra15] and provided the first example showing that submultiplicativity of rank can be strict, namelyThe analogous question for border rank, namely whether border rank can be strictly multiplicative under tensor product, remained open and we answer it in this paper. Specifically, we provide an example of a tensor T such that R(T ) = 5 and R(T ⊗T ) ≤ 24. We obtain Theorem 1.1. Border rank is not multiplicative under the tensor product.2010 Mathematics Subject Classification. 14M20, 15A69, 15A72.