ON THE TYPICAL RANK OF REAL POLYNOMIALS (OR SYMMETRIC TENSORS) WITH A FIXED BORDER RANK

Abstract

Let $\sigma _{b}(X_{m,d}(\mathbb {C}))(\mathbb {R})$ , $b(m+1) < {m+d \choose m}$ , denote the set of all degree d real homogeneous polynomials in m + 1 variables (i.e., real symmetric tensors of format (m + 1) × ⋯ × (m + 1), d times) which have border rank b over ℂ. It has a partition into manifolds of real dimension ≤ b(m + 1)−1 in which the real rank is constant. A typical rank of $\sigma _{b}(X_{m,d}(\mathbb {C}))(\mathbb {R})$ is a rank associated to an open part of dimension b(m + 1) − 1. Here, we classify all typical ranks when b ≤ 7 and d, m are not too small. For a larger set of (m, d, b), we prove that b and b + d − 2 are the two first typical ranks. In the case m = 1 (real bivariate polynomials), we prove that d (the maximal possible a priori value of the real rank) is a typical rank for every b.