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Daniel M. Pellegrino

Publications and source records attributed to Daniel M. Pellegrino.

15 recordsLinked to original sources
arXiv2026

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 estimate \[ D_m\le \sqrt m\,\exp\!\bigl(C\sqrt{\log m}\,\log\log m\bigr). \] The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: \[ D_{m,n_m}\longrightarrow 2 \qquad\text{whenever}\qquad \frac{n_m}{m}\longrightarrow1, \] and consequently \(\liminf_{m\to\infty}D_m\ge2\). Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.

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arXiv2026

Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm

The Kahane--Salem--Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:(\ell_\infty^n)^m\to\mathbb R$ satisfy \[ \|A\|\le C_m^{\mathrm{sym}}n^{(m+1)/2}. \] For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most $\sqrt{m\log m}\,\sqrt{m!}$; over the real scalar field, an elementary argument gives the sharper order $\sqrt m\,\sqrt{m!}$. We prove \[ C_m^{\mathrm{sym}}\ge \left(\sqrt{\frac2e}+o(1)\right)\frac{\sqrt{m!}}m, \] using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish \[ C_m^{\mathrm{sym}}\le C_0\sqrt{m!}, \] where $C_0$ is absolute, removing the factor $\sqrt m$ from the real upper bound. This estimate follows from a geometric argument in which rigidity for Gram permanents reduces the relevant configurations to sets controlled by Gaussian width. For unrestricted unimodular forms, we also obtain a quantitative rectangular estimate from truncated Hadamard matrices; in equal dimension, the normalized minimum is at most $1+o(1)$ whenever $m=o(n^{5/6})$.

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arXiv2026

Typical Bohnenblust--Hille Ratios

The polynomial Bohnenblust--Hille inequality controls the coefficient $\ell_{2m/(m+1)}$-norm of an $m$-homogeneous polynomial by its supremum norm, with a constant independent of the dimension. We study the associated Bohnenblust--Hille ratio from a probabilistic point of view, by placing normalized surface measure on the Euclidean coefficient sphere. Our results reveal a sharp contrast between the extremal behavior governing the classical Bohnenblust--Hille constants and the typical scale seen in coefficient directions. For arbitrary prescribed monomial supports, the ratio is eventually at most $1$ almost surely, and it tends to zero in spherical measure exactly when the number of monomials tends to infinity. On the full complex polynomial spaces we determine its typical asymptotic scale uniformly in the dimension; in the critical regime $n_m/m\to1$ this gives $2\sqrt2/\sqrt{m\log m}$, in contrast with the nonvanishing extremal scale. For real polynomials, the corresponding ratio tends to zero in spherical measure exactly when the number of variables tends to infinity.

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arXiv2026

Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality

The completely bounded Bohnenblust--Hille inequality of Arunachalam, Dutt, Escudero Gutiérrez and Palazuelos controls coefficient summability with optimal constant one, independently of the ambient dimension. We classify all nonzero equality cases over both the real and complex fields: they are precisely scalar multiples of Hadamard chains supported on equal-sided Cartesian boxes. Thus equality determines both the support and the factorization of the coefficient tensor. For each fixed degree $d\ge2$, we also establish sharp structural stability. A form with completely bounded norm one and deficit $\varepsilon$ lies within $C_d\sqrt\varepsilon$, in the critical coefficient $\ell_{2d/(d+1)}$ norm, of both a flat unimodular path and a normalized unitary chain on the same box. The corresponding unimodular and scaled unitary edges satisfy a matching normalized Frobenius estimate. At zero deficit both approximants coincide with the original Hadamard chain. All constants are independent of the ambient dimension, and the exponent $1/2$ is optimal for both approximations over either field.

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arXiv2026

Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials

We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on $\ell_p^n$. For $m$-homogeneous polynomials, the critical scale is $p=2m$: below this threshold no dimension-free comparison is possible, while at $p=2m$ we obtain $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.

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arXiv2026

The Geometry of Real Anisotropic Bohnenblust--Hille Constants

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust--Hille inequality. For an exponent vector $\mathbf q^{(m)}$, write $C_{\mathbf q^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m d_m/\log m\to\infty$; throughout this regime, their logarithm has the sharp scale $m d_m$. If also $d_m\to0$, then $\log C_{\mathbf q^{(m)}}^{(m)}/(m d_m)$ lies asymptotically in the interval $\left[\frac{\log 2}{4},\frac{2-\log 2-γ}{4}\right]$, where $γ$ is the Euler--Mascheroni constant; this interval has width less than $10^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)/4$ throughout a broad class of anisotropic regimes.

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arXiv2026

The Aron--Rueda zero-subspace problem

We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre--Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.

math.FA↗
arXiv2026

Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

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 $N$.

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arXiv2026

Hilbertian Kahane--Salem--Zygmund Inequalities: Extremizers and Quantitative Gaps

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 $r\leρ(n)$, where $ρ$ 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.

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arXiv2003

On ideals of polynomials and their applications

In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence theorems, Dvoretzky-Rogers and Extrapolation type theorems for dominated polynomials.

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arXiv2003

Cotype and nonlinear absolutely summing mappings

In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $% (2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from $\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$ has cotype $q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem.

math.FA↗
arXiv2003

Almost summing mappings

We introduce a general definition of almost $p$-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost $p$-summing multilinear mappings coincides with the whole space of continuous multilinear mappings.

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