On Maximum, Typical and Generic Ranks

Abstract

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.

v1: 8pp. v2: 9pp. Corrected gap in Theorem 6, other minor corrections. v3: 10pp. Minor changes