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Asymptotic contractivity of the Bohnenblust--Hille inequality for polynomials with few interacting variables

T0 review · 1 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper proves that for any fixed number of interacting variables M, the optimal Bohnenblust–Hille constant for m-homogeneous polynomials with at most M active variables tends to 1 as m grows.

desk verdict The main bound is new and the proof idea is good, but Eq. (11) is written with the wrong Fourier kernel: as stated Q_{σ,β}≡0, so the paper needs a small but essential fix before it is publishable. read the letter →

arxiv 2607.20847 v1 pith:JQRLKENL submitted 2026-07-23 math.FA

Jorge Nicolás Caro Montoya , Daniel Núñez Alarcón , Diana Marcela Serrano Rodríguez This is my paper · ORCID
classification math.FA MSC 46G2532A05
keywords Bohnenblust–Hilleinequalityhomogeneouspolynomialsoptimalconstantssparseinteractionsasymptoticcontractivitysupport-restrictedrandomcolouringParseval'sidentity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes asymptotic contractivity: for any fixed number of interacting variables M, the optimal Bohnenblust–Hille constant K_{m,M} on m-homogeneous polynomials supported on monomials with at most M active coordinates tends to 1 as the degree m grows. Concretely, 1 ≤ K_{m,M} ≤ A_M^{M/m} m^{(M^2−1)/(2m)} for an M-dependent constant A_M. This matters because the classical Bohnenblust–Hille constants grow with degree, and these results show that when interactions are few, the bounds become sharp. The argument is combinatorial: decompose by exact support size, project with bounded constants, randomly color the active variables, apply the multilinear inequality, then interpolate with Parseval's identity.

What carries the argument

The argument uses four ingredients: (i) a decomposition of a polynomial into exact support levels P_k with a dimension-free projection bound (Lemma 3.1); (ii) a random coloring of the n coordinates into k colors, so that monomials whose support receives all colors appear as coefficients of a k-linear form Q_{σ,β} with sup norm bounded by that of P; (iii) the classical multilinear Bohnenblust–Hille inequality applied to that form and summed over positive compositions β of m into k parts, yielding a degree exponent (k−1)(k+1)/(2k); (iv) interpolation between this ℓ^{p_M} bound and the ℓ^2 bound from Parseval to reach the final exponent.

What would settle it

Compute the optimal constants K_{m,2} for large m (say m = 20, 30, 40) using numerical convex optimization; if the values do not approach 1 but stay above 1 + ε for some ε > 0, the theorem's limit claim is false.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: for each fixed M, the optimal constant K_{m,M} satisfies 1 ≤ K_{m,M} ≤ A_M^{M/m} m^{(M^2−1)/(2m)}, and consequently K_{m,M} → 1. The proof bounds projections onto exact support levels, randomly colors the active variables to represent monomials as coefficients of a multilinear form, applies the classical multilinear Bohnenblust–Hille inequality, and interpolates the resulting ℓ^p estimate with Parseval's identity.

Load-bearing premise

The entire proof is over the complex scalar field; the norm identity z↦z^r mapping the polydisc onto itself fails over the reals, and the paper states that in the real case asymptotic contractivity already fails for M = 1.

Editorial extensions

If this is right

  • For every fixed M, the optimal constant K_{m,M} is asymptotically 1; the earlier progression from polynomial growth to uniform boundedness is sharpened to asymptotic contractivity.
  • The bound is quantitative: it gives a specific convergence rate K_{m,M} = 1 + O((log m)/m) for fixed M.
  • For M = 1, the paper proves the exact result K_{m,1} = 1 for all m, not just asymptotically.
  • The same qualitative conclusion follows from a more general support-sensitive inequality via interpolation, though with a larger power of m (M^2/(2m) instead of (M^2−1)/(2m)).
  • The proof is specialized to complex scalars; over the reals, asymptotic contractivity already fails for M = 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The random-coloring method likely extends to other ℓ^q coefficient norms by replacing Parseval with a general L^q endpoint, yielding analogous contractivity rates with modified interpolation exponents.
  • The homogeneous saving of 1/(2M) in the degree power is tied to the constraint that active exponents sum to m; in a nonhomogeneous setting the rate would be larger, as the paper notes.
  • The sharpness of the rate could be tested numerically for small M, e.g., by computing optimal constants for M = 2 and moderate m; the predicted exponent m^{3/(2m)} → 1 offers a concrete target.
  • If the complex assumption is essential, this points to a structural divide between real and complex polynomial geometry for sparse supports.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

Summary. The manuscript studies the optimal Bohnenblust–Hille constant K_{m,M} for m-homogeneous polynomials on C^n whose monomials involve at most M variables. The main result (Theorem 2.1) gives 1 ≤ K_{m,M} ≤ A_M^{M/m} m^{(M^2-1)/(2m)} for fixed M, so K_{m,M} → 1. The strategy is: split by support size, bound each exact-support level by a dimension-free constant using Bernoulli averaging, then use a random colouring/Fourier projection to reduce each level to the multilinear BH inequality, and finally interpolate with the ℓ^2 (Parseval) estimate.

Significance. If correct, the result is a clean asymptotic statement: on the M-sparse homogeneous class, the optimal BH constant is asymptotically 1 with explicit rate. The proof is combinatorial and self-contained (given classical BH), and the comparison with the recent support-sensitive inequality of [3] is useful context. The central argument is elegant and, apart from the sign error identified below, is sound.

major comments (1)
  1. [§4, Eq. (11)] Equation (11) defines Q_{σ,β} with kernel ω_1^{β_1}...ω_k^{β_k}. For a monomial z^α, the integrand contains ω_r^{∑_{j∈I_r} α_j + β_r}. Since β_r ≥ 1 and the sum is nonnegative, every ω_r appears with strictly positive exponent, so the normalized Haar integral over T^k vanishes. Thus Q_{σ,β} ≡ 0 and identity (14) is false. The projection actually selecting monomials with color-sum condition (13) is obtained with the conjugate kernel \overline{ω_1}^{β_1}...\overline{ω_k}^{β_k} (equivalently ω^{-β}). With that replacement, the exponent becomes S_r − β_r and (13) is exactly the selection condition; the rest of Proposition 4.1 and the subsequent averaging/counting argument are unaffected. This is a local but load-bearing correction.
minor comments (3)
  1. [§1, §4] The proof of (15) relies on z ↦ z^{β_r} mapping the complex unit disk onto itself. The authors state in §1 that the complex assumption is essential, but this should also be explicitly noted at the point of use in §4 so the reader sees where the scalar field enters.
  2. [§7] The application of [3, Theorem 2.12] is plausible, but the authors may want to verify the exact hypotheses of that theorem (e.g., whether the constant is c_1(q)^d or has a slightly different form) and adjust the displayed estimate accordingly.
  3. [General] Minor typesetting issues: several superscripts and subscripts are poorly rendered (e.g., ω1 β1, ℓpM), and the abstract contains an ungrammatical sentence fragment. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from independent, non-load-bearing external ingredients and explicit internal estimates.

full rationale

The derivation is self-contained and does not rely on the conclusion it proves. Theorem 2.1 is obtained by combining (i) Lemma 3.1, whose proof uses Bernoulli averaging and Lagrange interpolation to construct bounded projections with constants independent of the degree and dimension; (ii) Proposition 4.1, whose proof applies the classical multilinear Bohnenblust-Hille inequality, normalized Haar integration, and random colorings, and whose constants involve only B_k and the count of positive compositions; (iii) Theorem 5.1, which sums estimates over the exact support levels using monotonicity of l^p norms; and (iv) Section 6, which interpolates the support-level estimate with the trivial L^2 norm bound via Parseval's identity. No fitted parameter is renamed as a prediction, no equation is defined in terms of the quantity it is supposed to determine, and K_{m,M} is introduced as the optimal constant and then bounded from above by an independent expression. The citation of the support-sensitive inequality [3] in Section 7 is explicitly comparative and does not carry any weight in the proof of Theorem 2.1. The real-case restriction stated in Section 1 is a substantive limitation of the complex result, not a circular input. The proof's reliance on standard results such as the multilinear Bohnenblust-Hille inequality, Parseval's identity, and log-convexity of L^p norms is legitimate external support. A possible sign issue in the Fourier kernel of Eq. (11), if real, would be a correctness defect rather than an instance of the argument assuming its own target.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters. A_M is explicitly constructed in (23) from universal constants L_{M,k} and B_k; it is not fit to data. The only domain assumption is the complex scalar field, which is essential and stated by the authors.

assumptions (4)
  • domain assumption The scalar field is complex; z ↦ z^r maps D onto D (used in equation (15)).
    Essential for the norm identity between the polynomial Q_{\sigma,\beta} and its associated k-linear form; the authors note the real case fails for M = 1.
  • standard math Classical multilinear Bohnenblust–Hille inequality with dimension-free constants B_k (equation (8)).
    External theorem used in (16) to bound sums over random colour classes; constants depend only on k.
  • standard math Parseval's identity on T^n and log-convexity/interpolation of ℓ^p norms (equations (24)–(25)).
    Bridges the support-exponent p_M estimate to the target p_m estimate.
  • standard math Maximum modulus principle: sup norm on T^n equals sup norm on D^n (Section 7).
    Used only for comparison with [3, Theorem 2.12], not for the main proof.

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Pith. "Pith review of Asymptotic contractivity of the Bohnenblust--Hille inequality for polynomials with few interacting variables." pith.science (2026). https://pith.science/paper/JQRLKENL

@misc{pith2026260720847,
  author       = {Pith},
  title        = {Pith review of: Asymptotic contractivity of the Bohnenblust--Hille inequality for polynomials with few interacting variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQRLKENL}},
  note         = {Machine review of arXiv:2607.20847}
}
abstract

Let $K_{m,M}$ denote the optimal Bohnenblust--Hille constant on the class of $m$-homogeneous polynomials all of whose monomials involve at most $M$ different variables. We prove that, for every fixed $M$, these constants are asymptotically contractive: \[ \lim_{m\to\infty}K_{m,M}=1. \] More precisely, \[ 1\le K_{m,M}\le A_M^{M/m}m^{(M^2-1)/(2m)}, \] where $A_M$ depends only on $M$. The argument combines bounded projections onto exact support levels, a random colouring of the active variables, the classical multilinear Bohnenblust--Hille inequality and interpolation with Parseval's identity. We also point out that the qualitative conclusion follows from a recent, more general support-sensitive Bohnenblust--Hille inequality, although its direct application gives a slightly larger power of the homogeneous degree.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

    math.FA 2026-08 accept novelty 7.0 of 10

    For fixed q, the optimal constants BHint_{d,q} grow like exp(c_q sqrt(d log d)), which is subexponential, answering the question of Becker, Klein, Slote, Volberg and Zhang.

Reference graph

Works this paper leans on

4 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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    Defant, D

    A. Defant, D. Galicer, M. Mansilla, M. Mastyło and S. Muro,Support-sensitive Bohnenblust– Hille inequalities and local invariants on Hamming schemes, arXiv:2607.05594 (2026)

    arXiv 2026

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    H. F. Bohnenblust and E. Hille,On the absolute convergence of Dirichlet series, Ann. of Math. (2)32(1931), 600–622

    1931

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    Carando, A

    D. Carando, A. Defant and P. Sevilla-Peris,The Bohnenblust–Hille inequality combined with an inequality of Helson, Proc. Amer. Math. Soc.143(2015), no. 12, 5233–5238

    2015

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    M. Maia, T. Nogueira and D. Pellegrino,The Bohnenblust–Hille inequality for polynomials whose monomials have a uniformly bounded number of variables, Integral Equations Operator Theory88(2017), no. 1, 143–149. Departamento de Matemática Universidade Federal de Pernambuco 50740-560 - Recife, Brazil Email address:jorge.caro@ufpe.br Departamento de Matemátic...

    2017

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