On real typical ranks

TL;DR

This paper investigates the range of typical ranks for real tensors and polynomials, establishing which ranks are typical and providing specific results for low-dimensional cases like ternary cubics and quartics.

Contribution

It proves that all ranks between the minimal and maximal typical ranks are also typical and determines the specific typical ranks for certain low-dimensional symmetric tensors.

Findings

4 is the unique typical rank of real ternary cubics

Quaternary cubics have typical ranks 5 and 6

Ternary quartics have typical ranks 6, 7, and all are between 6 and 8

Abstract

We study typical ranks with respect to a real variety $X$. Examples of such are tensor rank ($X$ is the Segre variety) and symmetric tensor rank ($X$ is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of $n$-variate symmetric tensors of order $d$, or equivalently homogeneous polynomials of degree $d$ in $n$ variables, for small values of $n$ and $d$. We show that $4$ is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks $5$ and $6$ only. For ternary quartics we show that $6$ and $7$ are typical ranks and that all typical ranks are between $6$ and $8$. For ternary quintics we show that the typical ranks are between $7$ and $13$.