Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniformity for limits of tensors

Arthur Bik, Jan Draisma, Rob Eggermont, Andrew Snowden

Abstract

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank $\le r$ do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on $\mathbf{GL}$-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on $\mathbf{GL}$-varieties.

Uniformity for limits of tensors

Abstract

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank locus is not Zariski closed, and understanding the closure (the locus with "border rank" ) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on -varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on -varieties.
Paper Structure (40 sections, 44 theorems, 18 equations)

This paper contains 40 sections, 44 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Application to de-bordering
  3. $\mathbf{GL}$-varieties
  4. Existence of curves
  5. Image closures
  6. The Krull intersection theorem
  7. Overview of the proofs
  8. Connections to other work
  9. Notation
  10. Background on GL-varieties
  11. GL-varieties
  12. $\mathbf{GL}$-varieties as functors
  13. Shifting
  14. Decomposition theorem
  15. Chevalley's theorem
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $Z$ be as above. There is a function $\Phi \colon \mathbf{N} \to \mathbf{N}$, depending only on $Z$, such that $\mathop{\mathrm{brk}}\nolimits_Z(x) \le \Phi(\mathop{\mathrm{rk}}\nolimits_Z(x))$ for all $x \in \mathop{\mathrm{Sym}}\nolimits^d(V)$ and all finite dimensional $V$.

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: bdes
  • Theorem 2.3: bdes
  • Theorem 2.4: bdes
  • Theorem 3.1
  • Corollary 3.2
  • ...and 66 more