15.13.3 Tensor Symmetric Rank Minimality Condition

Tensor Symmetric Rank Minimality Condition is the requirement, central to the definition of a symmetric tensor's rank, that the number of pure power terms used in a decomposition be the smallest possible count achieving that decomposition, rather than merely any count for which a valid sum of pure power terms happens to reproduce the tensor. Because a symmetric tensor generally admits infinitely many decompositions into sums of pure power terms once a sufficiently large number of terms is allowed, the minimality condition is what turns an otherwise ambiguous notion, how many terms does it take, into a single well-defined invariant of the tensor, its symmetric rank.

Without the minimality condition, the term count of a decomposition would carry little information, since any decomposition can always be padded with additional terms that cancel or contribute nothing essential; imposing minimality forces attention onto the most economical representation, the one that most efficiently captures the tensor's true underlying complexity.

Formal Statement of the Condition

Rank as a Minimum Over All Decompositions

For a rank-n symmetric tensor T, its symmetric rank, denoted here r(T), is defined as the smallest integer r such that T can be written as a sum of r pure power terms:

where the minimum is taken over every possible choice of scalars c_k and vectors v_k achieving a valid decomposition of T.

Existence of a Finite Minimum

Because every symmetric tensor can be written as a finite linear combination of basis power terms, as guaranteed by the symmetric basis structure, the set over which this minimum is taken is always nonempty and bounded above by the dimension of Sym^n(V), so a finite minimal value always exists; the minimality condition selects the smallest member of this nonempty, bounded set of achievable term counts.

Why Minimality, Not Mere Existence, Defines Rank

Any Sufficiently Large Term Count Is Achievable

Given any decomposition of T using r terms, a decomposition using r plus one terms can always be constructed by adding a term that is itself zero, such as zero times some arbitrary vector's power, or by splitting one existing term into two terms whose sum reproduces the original; consequently, the mere existence of a decomposition using some particular number of terms conveys essentially no information about T unless that number is required to be minimal.

Rank as an Invariant Distinguishing Tensors

Because minimality forces attention onto the smallest achievable count, the symmetric rank becomes capable of distinguishing tensors of genuinely different structural complexity: a tensor that is itself a single pure power term has symmetric rank exactly one, satisfying the minimality condition trivially, while a tensor whose associated polynomial does not factor into fewer linear pieces requires a correspondingly larger minimal term count.

Difficulty of Verifying Minimality

Minimality Is Not Locally Checkable

Unlike many other properties encountered in symmetric tensor theory, confirming that a given decomposition into r terms is actually minimal cannot generally be done by inspecting that decomposition alone; it requires ruling out the existence of any decomposition using fewer terms, a global condition over the entire space of possible decompositions rather than a property verifiable from the r terms directly at hand.

Lower Bound Techniques

Establishing that a candidate decomposition is minimal typically proceeds by exhibiting an independent lower bound on the rank, often derived from properties of the associated polynomial or from algebraic invariants such as the rank of an associated matrix constructed from the tensor's components, and then matching that lower bound against an explicit decomposition achieving exactly that many terms, with the two together certifying minimality.

Minimality Within the Broader Rank Structure

The Condition as the Defining Feature of Symmetric Rank

The minimality condition is not an auxiliary technical detail but the defining feature that makes the notion of symmetric tensor rank meaningful and useful: every subsequent question about the rank structure, such as how rank behaves under the symmetric product or how it compares to the rank of the tensor viewed without the symmetry constraint, presupposes that rank has already been pinned down as this specific minimum.

Contrast With Upper-Bound-Only Notions

A related but weaker notion, simply asserting that T can be written using some bounded number of pure power terms without asserting minimality, provides an upper bound on the rank but not the rank itself; the minimality condition is precisely what upgrades such an upper bound into the exact value of the invariant under study.