Global Theory of Tensor Ranks

Abstract

The generic rank is considered under the complex number field, and it corresponds with the dimension of the secant variety. The dimension is studied in the area of algebraic geometry. In this chapter, we introduce known results and discuss the typical rank from the point of view of the Jacobian matrix. The generic rank attains the minimal typical rank. The typical rank of the set of tensors with format (m, n, p) for \(3\le m\le n\) is equal to \(\min (p,mn)\) if \(p>(m-1)n\). Note that the typical rank is sometimes not unique and the set of typical ranks consists of integers between some integers a and b. For a positive integer r, we consider the image of the summation map, which gives the set of tensors with rank less than or equal to r. Strassen and Tickteig introduced the idea of computing the dimension of the secant variety via the Jacobi criterion and the splitting technique. The generic rank of the set of tensors with format (m, n, p) is \(\lceil mnp/(m+n+p-2)\rceil \) if m, n, and p are sufficiently large integers, although the generic rank of the set of tensors with format (n, n, 3) is equal to \((3n+1)/2\) which is greater than \(\lceil 3n^2/(2n+1)\rceil \) if n is odd.