**On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank **

*Edoardo Ballico *

**Abstract**

Let
${\sigma}_{b}({X}_{m,d}(\mathbb{C}))(\mathbb{R})$,
$b(m+1)<\left(\genfrac{}{}{0ex}{}{m+d}{m}\right)$, 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*.