Polynomial growth of complex polynomial Bohnenblust--Hille constants

For complex \(m\)-homogeneous polynomials, let \(D_m\) denote the optimal dimension-free constant in the polynomial Bohnenblust--Hille inequality. We prove that, for every \(B>1/2\), there exists \(K_B>0\) such that \[ D_m\le K_B m^B,\qquad m\ge1. \] More precisely, we obtain the square-root-scale e