Symmetrization maps and minimal border rank Comon’s conjecture

Abstract

One of the fundamental open problems in the field of tensors is the border Comon’s conjecture: given a symmetric tensor F∈(Cn)⊗d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in (\mathbb {C}^n)^{\otimes d}$$\end{document} for d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document}, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in (Cn)⊗d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {C}^n)^{\otimes d}$$\end{document} of border rank n, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever n≤d+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\le d+1$$\end{document}. Our technical tools are border apolarity and border varieties of sums of powers.