Waring identifiable subspaces over finite fields

TL;DR

This paper introduces Waring identifiable subspaces over finite fields, exploring their classification and construction in relation to the Veronese variety, with applications to symmetric tensors and linear systems of quadrics.

Contribution

It defines and classifies Waring identifiable subspaces over finite fields, linking them to tensor decomposition and algebraic geometry.

Findings

Classification of Waring identifiable subspaces in ${ m P}^5({ m F}_q)$ and ${ m P}^9({ m F}_q)$

Construction methods for these subspaces

Applications to linear systems of quadrics in ${ m P}^3({ m F}_q)$

Abstract

Waring's problem, of expressing an integer as the sum of powers, has a very long history going back to the 17th century, and the problem has been studied in many different contexts. In this paper we introduce the notion of a Waring subspace and a Waring identifiable subspace with respect to a projective algebraic variety $\mathcal X$. When $\mathcal X$ is the Veronese variety, these subspaces play a fundamental role in the theory of symmetric tensors and are related to the Waring decomposition and Waring identifiability of symmetric tensors (homogeneous polynomials). We give several constructions and classification results of Waring identifiable subspaces with respect to the Veronese variety in ${\mathbb{P}}^5({\mathbb{F}}_q)$ and in ${\mathbb{P}}^{9}({\mathbb{F}}_q)$, and include some applications to the theory of linear systems of quadrics in ${\mathbb{P}}^3({\mathbb{F}}_q)$.