15.13.1 Tensor Symmetric Rank Decomposition Count

Tensor Symmetric Rank Decomposition Count is the actual numerical value r achieved by a minimal decomposition of a symmetric tensor into pure power terms, considered as a quantity to be computed, bounded, and compared across different tensors, rather than as the abstract minimality condition that defines what makes a particular count qualify as the rank in the first place. Where the minimality condition specifies the defining property a term count must satisfy to be called the rank, the decomposition count is the concrete number itself, together with the general bounds and special-case values that describe how large or small it can be for tensors of a given rank n over a given dimension d.

Studying the decomposition count as a quantity in its own right connects the abstract definition of symmetric rank to the concrete question most often asked in practice: given a specific symmetric tensor, or a specific rank n and dimension d, what values can this count take, and what typical or extreme behavior should be expected.

General Bounds on the Decomposition Count

Trivial Upper Bound From the Basis Expansion

Every symmetric tensor of rank n over a d-dimensional space can be written using at most C(d, n) pure power terms, simply by taking the basis expansion of the symmetric basis structure and observing that each basis element e^alpha is itself already a pure power term whenever alpha is concentrated entirely on a single index, or more generally by expressing a generic symmetric power combination as a sum of a bounded number of pure powers via polarization-style identities; this gives an upper bound on the decomposition count, though rarely the minimal one.

Lower Bound From Matrix Flattening

A standard lower bound on the decomposition count for a rank-n symmetric tensor is obtained by flattening the tensor into a matrix, pairing off some subset of its indices against the remainder, and computing the ordinary matrix rank of the result; because each pure power term contributes a rank-one matrix to any such flattening, the decomposition count must be at least as large as the matrix rank obtained from the most informative flattening.

Behavior at the Extremes

The Generic Decomposition Count

For most symmetric tensors of a given rank n over dimension d, meaning tensors chosen away from any special, lower-dimensional exceptional locus, the minimal decomposition count takes a single, predictable value known as the generic symmetric rank, which for many combinations of n and d is closely approximated by dividing the dimension of Sym^n(V) by the dimension of the space of pure power terms, though exceptions to this simple heuristic occur for certain specific combinations of n and d.

Special Tensors With Exceptionally Low Count

Certain symmetric tensors admit a decomposition count far smaller than the generic value, most notably tensors already expressible as a single pure power term, achieving the minimum possible count of one, or tensors lying on other special algebraic subvarieties of the space of symmetric tensors characterized by unusually efficient decompositions.

The Decomposition Count Under the Symmetric Product

Count for a Sum of Symmetric Tensors

If T and R are symmetric tensors of the same rank with decomposition counts r(T) and r(R) respectively, their sum T plus R has a decomposition count at most r(T) plus r(R), simply by concatenating a minimal decomposition of T with a minimal decomposition of R; this concatenated decomposition need not be minimal for the sum, so the true decomposition count of T plus R can be strictly smaller than this naive combined bound.

Count Under the Symmetric Product of Distinct Ranks

For the symmetric product T odot R of tensors of different rank, no simple additive relationship between the individual decomposition counts and the count of the product generally holds, since the pure power terms making up a decomposition of the product are not simply obtained by combining the pure power terms of the two factors; determining the decomposition count of such a product typically requires analysis specific to the tensors involved rather than a universal formula.

Computing the Decomposition Count in Practice

Small Cases Solvable by Direct Analysis

For low rank, such as n equal to two, the decomposition count coincides exactly with the ordinary matrix rank of the tensor's matrix representation, a quantity computable by standard linear algebra techniques such as Gaussian elimination or eigenvalue counting, making rank-two symmetric tensors the case in which the decomposition count is most straightforward to determine exactly.

Difficulty for Higher Rank

For rank three and above, determining the exact decomposition count of a specific given tensor is generally a substantially harder computational problem than the rank-two case, without a single universal algorithm analogous to Gaussian elimination, reflecting the genuine increase in structural complexity that accompanies higher-rank symmetric tensors relative to the well-understood rank-two, matrix-equivalent setting.