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

Abstract: 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.

• The paper improves the best known upper bound for the dimension-free constants from exp(O(√(m log m))) to exp((2.95+o(1))(log m)^2)) using fixed-ratio degree reduction, phase-preserving coefficient splitting, and central-mass capture.
• It proves the exact dimensional threshold D_{m,n_m}→1 if and only if n_m=o(m), showing that linear growth in the number of variables prevents asymptotic contractivity.
• It constructs entropy-rich polynomial witnesses that establish liminf D_m>1.27 and transfers the quasipolynomial degree dependence to polynomials valued in any fixed finite-dimensional complex Banach space.

The upper-bound problem and the main theorem

For an $m$-homogeneous polynomial $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$ on $\mathbb C^n$, the polynomial Bohnenblust–Hille inequality controls $\|a(P)\|_{q_m}$ by $\|P\|_\infty$, where $q_m=2m/(m+1)$ is the sharp critical exponent. Writing $D_{m,n}$ for the optimal $n$-dimensional constant and $D_m=\sup_n D_{m,n}$ for the dimension-free constant, the central question is the growth of $D_m$ in $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$0. Prior work had reduced the essentially exponential bound of order $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$1 to a hypercontractive exponential $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$2 [DFOOS] and then to the subexponential scale

$P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$3

[BPS]. The main result of this paper shows that this scale is not intrinsic:

$P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$4

i.e. $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$5, replacing $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$6 in the exponent by $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$7. A corollary places $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$8 below every stretched exponential: for all $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$9, $\mathbb C^n$0. The same quasipolynomial degree dependence transfers verbatim to vector-valued polynomials over any fixed finite-dimensional complex Banach space, via scalarization through the dual and absolute summing norms.

The paper also proves two lower-side results. First, an exact dimensional threshold: for any sequence $\mathbb C^n$1,

$\mathbb C^n$2

Second, the unrestricted constants are not asymptotically contractive; more precisely $\mathbb C^n$3. Thus the quasipolynomial upper bound coexists with a persistent noncontractive gap.

Fixed-ratio degree reduction with exact ancestry

The upper mechanism departs from the classical degree-reduction scheme of Bayart–Pellegrino–Seoane-Sepúlveda. That scheme reduces from degree $\mathbb C^n$4 to some $\mathbb C^n$5 via a factor $\mathbb C^n$6, and a rigidity proposition in the paper shows that at the critical scale $\mathbb C^n$7 this factor necessarily contributes $\mathbb C^n$8 with minimum leading coefficient $\mathbb C^n$9 — so iterating it cannot beat the $\|a(P)\|_{q_m}$0 barrier. The new argument instead keeps a fixed positive fraction of the degree.

Three ingredients combine into a one-step recurrence. First, a fractional two-block estimate: if $\|a(P)\|_{q_m}$1 is $\|a(P)\|_{q_m}$2-bihomogeneous with $\|a(P)\|_{q_m}$3, then

$\|a(P)\|_{q_m}$4

proved by combining Weissler's holomorphic hypercontractivity on each block with a Blei-type mixed-norm interpolation whose barycentric powers are dictated exactly by the identity $\|a(P)\|_{q_m}$5. Second, a phase-preserving splitting: each coordinate is independently mapped to $\|a(P)\|_{q_m}$6, $\|a(P)\|_{q_m}$7, or their average, and every child coefficient is indexed by a pair $\|a(P)\|_{q_m}$8 remembering its unique parent $\|a(P)\|_{q_m}$9. Consequently the retained $\|P\|_\infty$0 mass satisfies an exact identity before any triangle inequality is applied — no parent coefficients collide after absolute values are taken. Third, a uniform central-capture theorem: for every $\|P\|_\infty$1, a central window $\|P\|_\infty$2 retains at least $\|P\|_\infty$3 of the $\|P\|_\infty$4-mass of every parent coefficient, uniformly in dimension. The proof splits into a diffuse regime handled by Chebyshev under conditioning on no active split, and a dominant-coordinate regime handled by a half-mass principle applied to the symmetric unimodal sequence $\|P\|_\infty$5 along binomial expansion.

Together these yield, for the monotone envelope $\|P\|_\infty$6 (introduced because no monotonicity of $\|P\|_\infty$7 is known or used),

$\|P\|_\infty$8

Iterating this recurrence across geometrically decreasing degrees gives $\|P\|_\infty$9, and letting $q_m=2m/(m+1)$0 produces the golden-ratio constant above. The paper notes that the scheme is structurally analogous to fixed-scale improvement iterations in nonlinear elliptic regularity, with the degree playing the role of spatial scale; the polynomial loss accumulated per generation is precisely what generates the quadratic logarithm.

Critical dimension for asymptotic contractivity

The threshold $q_m=2m/(m+1)$1 is proved on both sides. In the subcritical regime, H\"older and Parseval give $q_m=2m/(m+1)$2, which tends to one whenever $q_m=2m/(m+1)$3. For the converse, a single Blaschke factor $q_m=2m/(m+1)$4 suffices: tensoring independent copies, truncating at total degree $q_m=2m/(m+1)$5, and homogenizing with one extra variable produces $q_m=2m/(m+1)$6-homogeneous witnesses in only $q_m=2m/(m+1)$7 variables, with label-preserving homogenization ensuring no coefficient merging. Since the radial growth index of $q_m=2m/(m+1)$8 is $q_m=2m/(m+1)$9, the construction yields

$D_{m,n}$0

with $D_{m,n}$1 the binary entropy function; the choice $D_{m,n}$2 gives the explicit benchmark $D_{m,n}$3. Because this obstruction already operates in linearly many variables, linear dimensional growth along even one subsequence precludes contractivity, and the threshold is exact. The paper emphasizes that fixed-interaction results do not imply this transition: even the support-sensitive estimate of Defant–Galicer–Mansilla–Mastyło–Muro recovers only the smaller regime $D_{m,n}$4.

A Gaussian benchmark clarifies why the extremal obstruction requires structure rather than abundance: a random polynomial with exponentially many coefficients and linearly many variables has Bohnenblust–Hille ratio tending to zero in probability, paying a $D_{m,n}$5 loss in the supremum norm. Persistent lower bounds require entropy generated while the supremum norm remains at the Hilbertian scale.

Entropy–radial principle and the certified witness

Normalizing squared coefficients as probabilities $D_{m,n}$6, the exact Rényi identity

$D_{m,n}$7

shows that deviation from the Hilbertian norm is measured precisely by Rényi entropy at the relevant order. For a rational inner seed $D_{m,n}$8 analytic beyond the closed polydisc, tensorization makes Shannon entropy additive while the radial index $D_{m,n}$9 measures truncation cost. The general principle reads

$n$0

so $n$1 is the efficiency ratio of the construction. The proof combines Cauchy estimates for exponentially small tail control, injectivity of the homogenization label map, and a conditional-entropy lemma showing that exponentially rare truncation preserves additivity, using finiteness of the second moment of the information variable (itself derived from Cauchy decay of Taylor coefficients).

Applying this to a certified bivariate rational-inner seed $n$2 with $n$3, $n$4 yields the stronger bound. Closed-bidisc stability is verified by an exact quadratic-form argument; the radial index is bounded rigorously by $n$5 via a boundary formula reducing positivity of a real quadratic form on $n$6 to exact rational checks. The entropy certificate $n$7 is obtained from a finite rational computation ($n$8 coefficient block, 300 terms of the arctanh series for $n$9), executed in SageMath 10.6 entirely in rational arithmetic; the margin over $D_m=\sup_n D_{m,n}$0 is about $D_m=\sup_n D_{m,n}$1. Combining the two certificates gives

$D_m=\sup_n D_{m,n}$2

the strongest explicit noncontractivity bound stated in the paper. This is the only computer-assisted step in the scalar theory, and it is an exact integer-arithmetic certificate rather than a floating-point test.

Secondary consequences

Two further sections port the same mechanisms. On the support-localized side, a direct colouring-and-composition recovery argument proves $D_m=\sup_n D_{m,n}$3 with $D_m=\sup_n D_{m,n}$4, recovering the sharp homogeneous exponent and implying $D_m=\sup_n D_{m,n}$5 whenever $D_m=\sup_n D_{m,n}$6 — a range the authors explicitly note was already known from prior work, so the contribution here is structural (label-preserving proof, quantitative auxiliary constants) rather than a new consequence. Conversely, multiplying the seed by $D_m=\sup_n D_{m,n}$7 enforces exact support $D_m=\sup_n D_{m,n}$8 with $D_m=\sup_n D_{m,n}$9 while preserving entropy, giving support-localized obstructions $D_m$0; for the certified witness this covers every density $D_m$1.

In the Pauli (noncommutative) setting, the paper records three concise results: the same critical-dimension criterion $D_m$2 with two-sided bounds $D_m$3; a support-adapted exact-recovery map into the Boolean cube via rank-one projections built from Pauli axes; and a tensor–entropy lower principle for homogeneous unitary seeds. These are applications of the two structural ideas, not a noncommutative analogue of the quasipolynomial theorem — indeed, the unrestricted noncommutative constants grow exponentially by recent work of Slote.

Limitations and open questions

Several limitations are stated plainly. The limsup constant $D_m$4 records the capture cost ($D_m$5 worst case) and the fixed contraction ratio, not a numerical optimization; improving either could sharpen it. The subcritical profile retains a single logarithmic gap between the lower scale $D_m$6 and the upper scale $D_m$7; closing it would require either showing that the unit ball cannot distribute entropy like the full combinatorial simplex, or constructing witnesses whose entropy per variable grows with $D_m$8 without radial cost — which the fixed-seed tensor construction cannot do, since its entropy density is fixed. In the interaction problem, the behavior for $D_m$9 is undetermined, and the paper argues this reflects a genuine gap between support-recovery cost and entropy production rather than slack in constants. Extending the fixed-ratio mechanism to infinite-dimensional Banach lattices would require a vector-valued fractional two-block estimate and is not pursued. Finally, whether the optimal leading constant in the subcritical regime equals a variational quantity such as $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$00 is left open, with no claim that this optimization is understood.

Conclusion

The paper replaces the previously best $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$01 upper bound for the complex polynomial Bohnenblust–Hille constants by a quasipolynomial $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$02 with an explicit limsup constant, via a fixed-ratio degree reduction that preserves exact coefficient ancestry through phase-preserving splitting and a uniform central-capture estimate. Complementarily, it establishes the exact linear dimensional threshold for asymptotic contractivity and a certified persistent gap $P(z)=\sum_{|\alpha|=m}a_\alpha z^\alpha$03 from organized, entropy-rich inner-function constructions. The remaining quantitative questions concern the subcritical rate, the intermediate interaction scale, and the optimal entropy density among admissible seeds.