Polynomials and tensors of bounded strength

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 generalised here to other tensors, is universal among these ranks in the following sense: any...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-08
Main Authors: Bik, Arthur, Draisma, Jan, Eggermont, Rob H
Format: Article
Language:English
Subjects:
Mathematical analysis
Polynomials
Strength
Tensors
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 generalised 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 generalises 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.
ISSN:2331-8422
DOI:10.48550/arxiv.1805.01816