Infinite-dimensional Waring rank and minimal decomposition

Determine whether an infinite-dimensional analogue of the finite-dimensional minimal Waring decomposition exists for symmetric functions of arbitrarily many variables, and thereby establish whether such a decomposition admits a representation with the smallest possible number of terms.

Background

The paper symmetrizes the lattice kinetic exponential over all permutations of lattice sites and suggests representing the resulting symmetric function as an infinite-dimensional Waring decomposition, namely, as a sum of products of one-variable functions. In finite dimensions, symmetric tensors admit minimal decompositions characterized by the Waring rank.

The paper explicitly notes that the functions and coefficients in such decompositions are generally nonunique, while the existence and characterization of a minimal representation in the infinite-dimensional setting are left unresolved. Resolving this issue would clarify the mathematical status and optimality of the proposed eigenfunction expansion.