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Border apolarity and varieties of sums of powers

Tomasz Mańdziuk, Emanuele Ventura

Abstract

We study border varieties of sums of powers ($\underline{\mathrm{VSP}}$'s for short), recently introduced by Buczyńska and Buczyński, parameterizing border rank decompositions of a point (e.g. of a tensor or a homogeneous polynomial) with respect to a smooth projective toric variety and living in the Haiman-Sturmfels multigraded Hilbert scheme. Their importance stems from the role of border tensor rank in theoretical computer science, especially in the estimation of the exponent of matrix multiplication, a fundamental and still unknown quantity in the theory of computation. We compare $\underline{\mathrm{VSP}}$'s to other well-known loci in the Hilbert scheme, parameterizing scheme-theoretic versions of decompositions. The latter ones are crucial in that they naturally explain the existing severe barriers to giving good lower bounds on ranks. We introduce the notion of border identifiability and provide sufficient criteria for its appearance, which rely on the multigraded regularity of Maclagan and Smith. We link border identifiability to wildness of points. Finally, we determine $\underline{\mathrm{VSP}}$'s in several instances and regimes, in the contexts of tensors and homogeneous polynomials. These include concise $3$-tensors of minimal border rank and in particular of border rank three.

Border apolarity and varieties of sums of powers

Abstract

We study border varieties of sums of powers ('s for short), recently introduced by Buczyńska and Buczyński, parameterizing border rank decompositions of a point (e.g. of a tensor or a homogeneous polynomial) with respect to a smooth projective toric variety and living in the Haiman-Sturmfels multigraded Hilbert scheme. Their importance stems from the role of border tensor rank in theoretical computer science, especially in the estimation of the exponent of matrix multiplication, a fundamental and still unknown quantity in the theory of computation. We compare 's to other well-known loci in the Hilbert scheme, parameterizing scheme-theoretic versions of decompositions. The latter ones are crucial in that they naturally explain the existing severe barriers to giving good lower bounds on ranks. We introduce the notion of border identifiability and provide sufficient criteria for its appearance, which rely on the multigraded regularity of Maclagan and Smith. We link border identifiability to wildness of points. Finally, we determine 's in several instances and regimes, in the contexts of tensors and homogeneous polynomials. These include concise -tensors of minimal border rank and in particular of border rank three.
Paper Structure (33 sections, 81 theorems, 88 equations, 1 table)

This paper contains 33 sections, 81 theorems, 88 equations, 1 table.

Table of Contents

  1. Introduction
  2. Cox rings, ranks and border apolarity
  3. Cox rings and Picard groups
  4. Toric apolarity and ranks
  5. Multigraded Hilbert schemes and border apolarity
  6. Multigraded regularity and Hilbert schemes of toric varieties
  7. Maclagan-Smith construction and the map to the Hilbert scheme
  8. The morphism from $\mathrm{Slip}_{r,X}$ to $\mathrm{Hilb}_{sm}^r(X)$
  9. Limiting schemes
  10. $\underline{\mathrm{VSP}}$ versus $\mathrm{VSP}$ versus $\mathrm{VPS}$
  11. Basic definitions and first properties
  12. Birational results
  13. Membership in $\mathrm{Slip}_{r,\mathbb{P}^n}$, complete intersections and monomials
  14. Ext criterion for complete intersections
  15. Applications to $\underline{\mathrm{VSP}}$'s of monomials
  16. ...and 18 more sections

Key Result

Theorem 1

Let $F\in T_\mathbf{v}$ be a homogeneous polynomial of degree $\mathbf{v}$ and $r$ be a positive integer. Then $\phi_{r,X}$ induces a bijection between the set of those irreducible components of the closure of $\mathrm{VPS}(F,r)\cap \mathrm{Hilb}_{sm}^r(X)$ that contain a scheme with the generic Hil

Theorems & Definitions (188)

  • Theorem : Theorem \ref{['birmap1']}
  • Theorem : Theorem \ref{['prop:complete_intersection']}
  • Theorem : Theorem \ref{['thm:idenfiability via toric regularity']}
  • Theorem : Theorem \ref{['thm:minimalBRTensors']}
  • Theorem : Theorem \ref{['theo:min border rank three']}
  • Example 2.1
  • Definition 2.2: Nef Cone
  • Example 2.3
  • Example 2.4
  • Definition 2.5: Apolar ideals
  • ...and 178 more