## Abstract

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalized here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalizes a theorem by Derksen-Eggermont-Snowden on cubic polynomials, as well as a theorem by Kazhdan-Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.

Original language | English |
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Article number | 1850062 |

Number of pages | 19 |

Journal | Communications in Contemporary Mathematics |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Nov 2019 |

## Keywords

- invariant theory
- Noetherianity up to symmetry
- polynomial functors
- strength
- Tensor rank
- Waring rank