Finiteness of Hadamard ranks

TL;DR

This paper classifies projective varieties with finite Hadamard rank for all points and establishes bounds on the maximum Hadamard rank for specific algebraic varieties.

Contribution

It provides a classification of varieties with universally finite Hadamard rank and sharp bounds for maximum Hadamard rank in certain cases.

Findings

Classified varieties with finite Hadamard rank for any point.

Proved finiteness of Hadamard rank for tensor-related varieties.

Established sharp upper bounds on Hadamard rank for specific families.

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.