Algebraic and computational aspects of real tensor ranks (Q267720)

Let \(T\) be a tensor of format \(n_1\times \cdots \times n_k\) over a field \(K\). The rank of \(T\) over \(K\) is the minimal number of summands in a decomposition of \(T\) as a sum (with coefficients in \(K\)) of rank \(1\) tensors. In the monograph under review the field is usually \(\mathbb {R}\), but the case \(\mathbb {C}\) is also needed as a tool (the real rank of a real tensor may be higher than its complex rank). Among the tensors of fixed format the generic rank is the one obtained (for complex tensors) on a non-empty Zariski open subset of the complex space of tensors and it is known that the value of the generic tensor is related to the dimensions of the secant varieties of Segre embeddings of multiprojective space; here the literature is quoted, in particular the value of the generic rank for \(2\times \cdots \times 2\) tensors proved in [\textit{M. V. Catalisano} et al., J. Algebr. Geom. 20, No. 2, 295--327 (2011; Zbl 1217.14039)]. For real tensors the typical ranks are all the ranks realized by a non-empty Euclidean open subsets of the real tensor. The minimal typical rank is the generic one, while the maximal is at most twice the minimal one [\textit{G. Blekherman} and \textit{Z. Teitler}, Math. Ann. 362, No. 3--4, 1021--1031 (2015; Zbl 1326.15034)]. This monograph starts with the basics (motivations, applications, flattening). Most chapters deal with 3-tensors (\(k=3\)), with several results on ranks (maximal and typicals) proved by the authors in recent papers, e.g., [Linear Multilinear Algebra 63, No. 5, 940--955 (2015; Zbl 1310.15043); ibid. 438, No. 2, 953--958 (2013; Zbl 1260.15042)]. One chapter (Absolutely non-singular tensors and determinantal polynomials), looks at \(3\)-tensors \(T\) of type \(u\times n\times m\), \(u\geq n\), given by \(m\) \(u\times n\) matrices \(M_1,\dots ,M_m\) all whose real combinantions (except the zero one) have rank \(n\) (they search for the maximal \(m\) for fixed \(u, n\) for which some \(T\) exists; when \(u=n\) this maximal value is known and it involves the Hurwitz-Radon numbers).