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Finiteness of Hadamard ranks

Dario Antolini, Edoardo Ballico, Alessandro Oneto

Abstract

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard rank is finite for any point. As a by-product we obtain the finiteness of the Hadamard rank with respect to varieties of tensors, such as Grassmannians, Chow varieties, varieties of reducible forms and their secant varieties, complementing previous known results on secant varieties of Segre-Veronese varieties. We prove sharp upper bounds on the maximum Hadamard rank for certain families of algebraic varieties: this is a consequence of a result on the lower semi-continuity of the Hadamard rank for curves that do not contain points with at least two zero coordinates.

Finiteness of Hadamard ranks

Abstract

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard rank is finite for any point. As a by-product we obtain the finiteness of the Hadamard rank with respect to varieties of tensors, such as Grassmannians, Chow varieties, varieties of reducible forms and their secant varieties, complementing previous known results on secant varieties of Segre-Veronese varieties. We prove sharp upper bounds on the maximum Hadamard rank for certain families of algebraic varieties: this is a consequence of a result on the lower semi-continuity of the Hadamard rank for curves that do not contain points with at least two zero coordinates.

Paper Structure

This paper contains 8 sections, 12 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Hadamard products of algebraic varieties
  3. Finiteness of the maximum rank
  4. Varieties of tensors
  5. Grassmannians
  6. Tangential variety to Veronese varieties
  7. Chow varieties and varieties of reducible forms
  8. Border Hadamard rank

Key Result

Theorem A

The Hadamard-$X$-rank is finite for every point if and only if $X$ is strongly concise, namely, for any $i \in \{0,\ldots,N\}$, $(X \cap H_i) \not\subset \bigcup_{j\neq i} H_j$.

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Example 2.3: Conciseness
  • Example 2.4
  • Example 2.5
  • Theorem 2.6: hadamardranks
  • ...and 36 more