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

TL;DR

This paper classifies typical ranks of real symmetric tensors with fixed border rank, revealing the structure of their rank distributions for various parameters and establishing new results for bivariate cases.

Contribution

It provides a complete classification of typical ranks for certain border ranks and degrees, and characterizes the rank behavior in the case of real bivariate polynomials.

Findings

Classifies all typical ranks for b ≤ 7 and moderate m, d.

Proves b and b+d-2 are the first typical ranks for larger parameter sets.

Shows that for m=1, the maximal real rank d is always typical.

Abstract

Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ... \times (m+1)$, $d$ times) which have border rank $b$ over $\mathbb {C}$. It has a partition into manifolds of real dimension $\le b(m+1)-1$ 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 $b\le 7$ and $d, m$ are not too small. For a larger sets 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 $b$.