Weak Hadamard Star Configurations and Apolarity

TL;DR

This paper generalizes Hadamard star configurations in projective space, classifies weak Hadamard star configurations, and explores their apolarity properties with respect to generic homogeneous polynomials.

Contribution

It introduces weak Hadamard star configurations, extends previous constructions, and studies their apolarity in the case when codimension equals the dimension.

Findings

Classification of weak Hadamard star configurations

Existence results for apolar configurations when c=n

Extension of Hadamard star configuration theory

Abstract

In [7] the authors have introduced a new construction using the Hadamard product to present star configurations of codimension $ c $ of $\mathbb{P}^n$ and which they called Hadamard star configurations. In this paper, we introduce a more general type of Hadamard star configuration. Any star configuration constructed by our approach is called a weak Hadamard star configuration. We classify weak Hadamard star configurations, and in the case $ c=n $, we investigate the existence of a (weak) Hadamard star configuration apolar to the generic homogeneous polynomials of degree $ d$.