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- W2951318069 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 $ble 7$ 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 $b$." @default.
- W2951318069 created "2019-06-27" @default.
- W2951318069 creator A5080462516 @default.
- W2951318069 date "2013-07-09" @default.
- W2951318069 modified "2023-09-27" @default.
- W2951318069 title "On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank" @default.
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