On apolarity and generic canonical forms
The aim of this paper is to generalize the theory of apolarity and generic canonical forms. A vector space \(A\) together with a set \(\mathcal S\) of multilinear forms on the space \(A\) is called an \(\mathcal S\)-algebra. The \(\mathcal S\)-algebra is a generalization of the notion of an algebra. Let \(A\{x_1,\dots,x_d\}\) be the space of all functions on the space \(A\) to itself that may be constructed by the multilinear forms in the set \(\mathcal S\). These functions may be considered as polynomials. Let \(V\) and \(W\) be vector spaces and let \(f\) be a linear map from \(W\) to \(V\). A linear function \(L\in V^*\) is said to be apolar to \(f\) if \(\langle L| f(w)\rangle=0\) for all \(w\in W\). The main theorem states roughly that a form \(p\in A\{x_1,\dots,x_d\}\) in an \(\mathcal S\)-algebra \(A\) is canonical if and only if there exist some linear maps generated by a set of vectors in \(A\) such that no nonzero dual element is apolar to all these linear maps. As an application of the theory, the author studies the rank of general tensors, symmetric and skew-symmetric tensors. The problem of finding upper bounds for the essential rank of general tensors has a combinatorial interpretation. An upper bound for the number of rooks required to cover a multidimensional chess board is also an upper bound for the essential rank of tensors. In this paper, similar combinatorial bounds are found for symmetric and skew-symmetric tensors. With the help of Steiner triple systems, explicit bounds for the essential rank of symmetric and skew-symmetric tensors of degree 3 are found.
scientific article; zbMATH DE number 1270879
scientific article