Typical Ranks

Abstract

Let m and n be positive integers with $m\le n$. If one chooses an $m\times n$ matrix M randomly according to a distribution whose probability density function is positive anywhere, then almost always $\mathrm {rank}M=m$. Moreover, the maximal rank of $m\times n$ matrices is m. Thus, in the matrix case, i.e., the 2-tensor case, if one chooses a matrix with specified size randomly according to a distribution whose probability density function is positive anywhere, its rank is almost always the maximal rank of the specified size. The phenomenon is quite different in the case of 3 or higher dimensional tensors. Let m, n, and p be positive integers. There may be multiple integers r such that the set of rank r m $\times$ n $\times$ p tensors over $\mathbb {R}$ contains a nonempty Euclidean open subset. Thus, there may be multiple integers r such that if one chooses an $m\times n\times p$ tensor randomly according to a distribution whose probability density function is positive anywhere, then the probability that the chosen tensor has rank r is positive. These integers are called the typical ranks of $m\times n\times p$ tensors over $\mathbb {R}$. If the base field is algebraically closed, e.g., $\mathbb {C}$, there is an integer r such that if one chooses an $m\times n\times p$ tensor randomly according to a distribution whose probability density function is positive anywhere, then almost always, the rank of the chosen tensor is r. This integer is called the generic rank of $m\times n\times p$ tensors over $\mathbb {C}$. In many cases, the generic rank of $m\times n\times p$ tensors over $\mathbb {C}$ is strictly less than the maximal rank of $m\times n\times p$ tensors over $\mathbb {C}$. If the base field is $\mathbb {R}$, the phenomenon is more complicated. In this chapter, we discuss typical ranks over $\mathbb {R}$ and the generic rank over $\mathbb {C}$. For the readers’ convenience, some basic facts of algebraic geometry are included.