Agler class membership of symmetric multiaffine stable rational inner functions

Determine whether, for any symmetric multiaffine polynomial p(z_1, …, z_d) with no zeros in D^d, the associated rational inner function φ(z) = \tilde{p}(z)/p(z) necessarily belongs to the Schur–Agler class.

Background

The Schur–Agler class consists of bounded analytic functions on D^d that satisfy the operator inequality |f(T)| ≤ 1 for all commuting strictly contractive d-tuples T. In dimensions d ≥ 3, not all bounded analytic functions (nor all rational inner functions) satisfy this inequality, due to known counterexamples to von Neumann/Andô-type inequalities.

The paper provides sufficient, checkable conditions in prior work for certain symmetric multiaffine polynomials p with no zeros in D^d to yield φ = \tilde{p}/p in the Schur–Agler class. However, it remains unknown whether symmetry and multiaffinity alone guarantee φ ∈ Schur–Agler, which would strengthen classical results such as the Grace–Walsh–Szegő theorem.