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

Abstract

Let ${\sigma }_b(X_{m,d}(\mathbb{C}))(\mathbb{R})$, $b(m+1)<\left(\genfrac{}{}{0ex}{}{m+d}{m}\right)$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ...\times (m+1)$, $d$ 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 $b$.

Acta Mathematica Vietnaminica (to appear)

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