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

Waring identifiable subspaces over finite fields