On quasipolynomial upper bounds for complex polynomial Bohnenblust--Hille constants

For an m-homogeneous polynomial on C^n, let D_{m,n} denote the optimal constant in the complex polynomial Bohnenblust–Hille inequality, and set D_m = sup_n D_{m,n}. Our principal result gives a quasipolynomial upper bound for the dimension-free constants: limsup_{m->infinity} log D_m / (log m)^2 <= 5 / [8 log(4/(1+sqrt(5)))]. Thus the previously known exp(O(sqrt(m log m))) bound can be replaced by exp(O((log m)^2)). The proof uses a fixed-ratio degree reduction. A phase-preserving splitting retains the exact ancestry of every coefficient, and a fractional two-block estimate controls the two resulting degree scales. This gives a recurrence between macroscopically separated degrees whose iteration produces the quadratic logarithm above. We also obtain lower bounds from tensorized inner functions and entropy–radial control. This gives localized estimates and, for a certified bivariate rational-inner seed, liminf_{m->infinity} D_m > 1.27. Thus the new quasipolynomial upper bound coexists with a persistent noncontractive gap.