15.14.2 Tensor Symmetric Decomposition Pure Power Form

Tensor Symmetric Decomposition Pure Power Form is the name given to the individual building block used in a symmetric tensor decomposition: a rank-one symmetric tensor obtained by taking the tensor power of a single vector with itself repeatedly, rather than combining several distinct vectors. Every term appearing in a symmetric decomposition is required to be of this pure power shape, and the Pure Power Form is the concept that isolates and studies this shape on its own, independently of how many such terms are summed together.

Definition

The Pure Power Construction

Given a vector v in a vector space V and an order d, the pure power form associated with v is the symmetric tensor obtained by tensoring v with itself d times:

with v repeated d times. This tensor is automatically symmetric, since permuting the factors of a repeated tensor product of the same vector leaves the product unchanged.

Coordinate Description and Polynomial Correspondence

Under the standard identification of symmetric tensors of order d with homogeneous polynomials of degree d, the pure power form of a vector corresponds precisely to the d-th power of a single linear form. If the vector v has coordinates that define a linear form L, the corresponding pure power form is the polynomial obtained by raising L to the d-th power. This correspondence is the reason pure power forms are also referred to as powers of linear forms, and it is the basic dictionary connecting tensor decomposition to the classical algebraic problem of writing forms as sums of powers of linear forms, historically known as the Waring problem for forms.

Structural Properties

Rank-One Status

A pure power form is, by construction, a rank-one element both within the space of symmetric tensors and within the ambient space of all tensors of that order: no proper sum of two or more nonzero terms is needed to express it, and its symmetric rank and ordinary rank both equal one. Consequently, the pure power forms are exactly the points of the affine cone over the Veronese variety, and their projectivizations are exactly the points of the Veronese variety itself.

Uniqueness of the Generating Vector

Two vectors produce the same pure power form, for order d greater than or equal to two, precisely when one is obtained from the other by multiplying by a d-th root of unity. Over the real numbers with even order, this reduces further to the two vectors being equal up to an overall sign. This near-uniqueness of the generating vector is what allows the pure power form to serve as an unambiguous coordinate for a point of the Veronese variety, once the residual root-of-unity action is quotiented out.

Distinguishing Pure Power Forms from General Rank-One Tensors

Within the ambient space of all tensors of order d (not restricted to symmetric ones), the rank-one tensors are of the more general shape given by tensoring together d possibly distinct vectors. The Pure Power Form is the special case in which all d vectors coincide. This distinction underlies the structural gap between the Veronese variety, which parametrizes pure power forms, and the larger Segre variety, which parametrizes all rank-one tensors, and it is the geometric seed from which the difference between symmetric rank and ordinary rank, discussed under the Rank Relation, ultimately grows.

Role in Decomposition Theory

Building Block of Symmetric Decompositions

Every symmetric decomposition of a tensor T expresses T as a sum of pure power forms:

so that the theory of symmetric decomposition can be phrased entirely as the theory of expressing symmetric tensors as sums of pure power forms, and the minimal number of pure power forms required is exactly the symmetric rank.

Waring Decomposition Terminology

When the ambient object is regarded as a homogeneous polynomial rather than a tensor, a decomposition into pure power forms is traditionally called a Waring decomposition of the polynomial, and the minimal number of powers of linear forms needed is called the Waring rank. The Pure Power Form concept is thus the tensor-theoretic incarnation of the "power of a linear form" appearing in the classical Waring problem, and results about Waring rank translate directly into results about symmetric tensor rank through this dictionary.

Apolarity and Pure Power Forms

The apolarity pairing between forms and differential operators is especially simple on pure power forms: a linear differential operator dual to a vector w annihilates the d-th power of the form dual to v to an order controlled by whether w is proportional to v, and this simple annihilation behavior is what makes apolar ideals and catalecticant matrices effective tools for detecting how a given tensor decomposes into pure power forms.

Geometric Picture

The Veronese Variety as the Locus of Pure Power Forms

The set of all pure power forms, up to scalar, forms the Veronese variety inside the projective space of symmetric tensors, obtained as the image of the projective space of vectors under the Veronese embedding, which sends a point to its d-th power. The dimension, degree, and secant behavior of this variety are the geometric data that determine the possible symmetric ranks, the expected generic symmetric rank, and the classification of exceptional cases described by the Alexander-Hirschowitz theorem.