Variational optimization of rational-inner entropy efficiency

Determine the variational quantities governing the optimal entropy efficiency of admissible rational inner seeds, namely the supremum of h(Φ)/(2d) over nonconstant rational inner functions on the polydisc and the analogous optimization of h(Φ)/(2Λ(Φ)), and establish whether either variational problem is understood.

Background

The entropy–radial lower-bound construction associates each admissible rational inner seed Φ on the d-dimensional polydisc with a Shannon coefficient entropy h(Φ), dimension d, and radial index Λ(Φ). The resulting lower bounds depend on the ratios h(Φ)/(2d) in dimension-localized settings and h(Φ)/(2Λ(Φ)) for unrestricted constants.

The paper explicitly notes that these quantities naturally define optimization problems over rational inner functions but does not claim that either problem is understood. Resolving them would determine how efficient the paper’s tensorized inner-function constructions can be and could improve the lower bounds for polynomial Bohnenblust–Hille constants.

References

One may therefore consider

\mathcal H_d:= \sup_{\Phi}\frac{h(\Phi)}{2d},

where the supremum ranges over nonconstant rational inner functions on $\mathbb Dd$ analytic beyond the closed polydisc, or the analogous radial efficiency $h(\Phi)/(2\Lambda(\Phi))$. We do not claim that either variational problem is understood; the notation merely records the optimization naturally suggested by the proof.

— On quasipolynomial upper bounds for complex polynomial Bohnenblust--Hille constants  (2608.16584 - Pellegrino et al., 17 Aug 2026) in Remark 2.??, Section 5.2, “A seed variational quantity”