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.
D. Pellegrino, Eduardo V. Teixeira· 2 citations· ⚡2
We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of order $n$ exists and $r\le\rho(n)$, where $\rho$ is the Hurwitz--Radon function. If equality fails, we obtain an explicit gap above $\sqrt n$. We also prove two asymptotic results. If $1\le m_n\le n$ and $\limsup r_n/\log_2 n<2$, there are sign forms on $\ell_2^{r_n}\times\ell_2^{m_n}\times\ell_2^n$ with norm $(1+o(1))\sqrt n$. In the square case, if $r\ge2\lceil\log_2(8n)\rceil$, then $K_{r,n}-1\ge c(1+\log_2 n)^{-4}$. We also prove a fourth-moment estimate in every fixed multilinear order, characterize equality, and give exact and asymptotic constructions.
Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust--Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.
D. Pellegrino, A. Raposo· 2 citations· ⚡1
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