Orbit compression and asymptotic contractivity for symmetric Bohnenblust--Hille inequalities

Abstract

For a symmetric complex m-linear form, passage to the diagonal polynomial aggregates entire permutation orbits of ordered coefficients. We show that the multiplicities of these orbits, usually discarded in a polarization argument, determine the asymptotic size of the corresponding Bohnenblust--Hille constants. More precisely, the optimal symmetric constant is bounded by 1+C/m for all sufficiently large m; the same estimate holds with the stronger diagonal-polynomial norm on the right-hand side. The argument rests on an exact weighted orbit identity, a support-sensitive coefficient estimate, and a cutoff separating the low- and high-support regimes. We further prove asymptotic contractivity uniformly on the full critical anisotropic face, with bound 1+C/sqrt(m), and obtain the rate 1+C_{s,delta}/m for forms invariant under a fixed number of nondegenerate symmetry blocks. Finally, a family of real symmetric forms with Bohnenblust--Hille ratio sqrt(2) in every even degree shows that the phenomenon is intrinsically complex.