Polynomial growth of complex polynomial Bohnenblust--Hille constants

Abstract

For an $m$-homogeneous polynomial on $\mathbb C^n$, let $D_{m,n}$ denote the optimal constant in the complex polynomial Bohnenblust--Hille inequality, and set $D_m:=\sup_{n\ge1}D_{m,n}$. We prove that the dimension-free constants $(D_m)$ have at most polynomial growth: there are absolute constants $B_0,K<\infty$ such that \[ D_m\le K m^{B_0} \qquad(m\ge1). \] This replaces the previously best general estimate \[ D_m\le \exp\!\bigl(O(\sqrt{m\log m})\bigr) \] by a fixed power of the degree---a qualitative change in the known growth scale. The proof has two stages. A phase-preserving fixed-ratio decomposition retains the exact ancestry of every coefficient and first yields an explicit quasipolynomial estimate. A weighted graded bootstrap then prevents the one-step loss from accumulating: balanced degree splits produce a strict binary-entropy contraction, while dominant powers are isolated by contractive spectral projections and compressed isometrically to lower degree. This proves polynomial growth without optimizing the exponent. A sharper analysis of the same architecture yields $D_m=o(m^\mu)$ for every $\mu>\beta_\star$, where $\beta_\star<2.47$ is the sharp threshold of the present two-regime bootstrap. On the lower side, we prove the sharp dimensional criterion \[ D_{m,n_m}\longrightarrow1 \quad\Longleftrightarrow\quad n_m=o(m), \] together with the certified estimate \[ \liminf_{m\to\infty}D_m>1.27. \] As an application, the polynomial bound yields an explicit logarithmic remainder in the multidimensional Bohr-radius asymptotic.