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The polarization constant of finite dimensional complex spaces is one

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Pith's one-line read For every finite-dimensional complex Banach space, the polarization constant equals 1.

desk verdict The main theorem is correct and settles the natural open problem: every finite-dimensional complex Banach space has polarization constant 1, with a clean proof and worthwhile extras on the real case and nuclear norms. read the letter →

arxiv 1908.08107 v2 pith:Z5PV4OQO submitted 2019-08-21 math.FA math.CV

Verónica Dimant , Daniel Galicer , Jorge Tomás Rodríguez This is my paper · ORCID
classification math.FAmath.CV MSC 46G2547A0715A6946T25
keywords polarizationconstanthomogeneouspolynomialssymmetricmultilinearformsfinite-dimensionalcomplexBanachspaceholomorphicfunctionscomplexificationofrealspacestypeandcotypenuclearnorm
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 proves that in any finite-dimensional complex Banach space, the polarization constant $\mathbf{c}(X)$ is exactly $1$. This means that for high-degree homogeneous polynomials, the norm of the associated symmetric multilinear form is asymptotically the same as the polynomial's uniform norm. The value $1$ is the smallest possible, so the result is sharp and it matches what was already known for inner-product spaces. The paper also shows the real analogue fails, with some real spaces having polarization constant strictly above $1$, and it connects the real case to a complexification procedure.

What carries the argument

The load-bearing object is the exact formula for the polarization constants of finite-dimensional complex $\ell_1$ spaces, which reduces the constant to a maximum over multi-indices $k_1+\cdots+k_d=k$ of $$\frac{k_1!\cdots k_d!}{k!}\,\frac{k^k}{$k_1^{{k_1}}$\cdots $k_d^{{k_d}}$}.$$ That maximum is attained at the balanced multi-index, and its $k$-th root growth is $1/e$ multiplied by $(k^k/k!)^{1/k}$, which tends to $e$, so the product tends to $1$. Around this, an approximate-quotient lemma shows every finite-dimensional complex space can be replaced by $\ell^d_1(\mathbb{C})$ at multiplicative cost $(1+\varepsilon)$ per factor, and the balanced-maximum computation closes the limit.

What would settle it

Compute $\mathbf{c}(k,\ell^2_1(\mathbb{C}))$ for large $k$ by evaluating the displayed maximum over $k_1+k_2=k$; if $(\mathbf{c}(k,\ell^2_1(\mathbb{C})))^{1/k}$ does not tend to $1$, Proposition 2.1 and hence Theorem 1.1 fail. More directly, any finite-dimensional complex space with a sequence of norm-one polynomials $P_k$ satisfying $\|P_k^{\vee}\|^{1/k}>1+\delta$ for some $\delta>0$ would refute the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any finite-dimensional complex Banach space $X$, $\mathbf{c}(X)=1$, where $\mathbf{c}(X)=\limsup_{k\to\infty}\mathbf{c}(k,X)^{1/k}$. The proof first settles the extremal case $X=\ell^d_1(\mathbb{C})$ using the exact formula displayed as equation (9) for $\mathbf{c}(k,\ell^d_1(\mathbb{C}))$; a Stirling and arithmetic-geometric mean estimate shows the $k$-th root tends to $1$. It then shows every finite-dimensional complex $X$ is an almost-isometric quotient of $\ell^d_1(\mathbb{C})$: a norm-one surjection $q$ with every unit vector $x$ admitting a preimage of norm $<1+\varepsilon$. Pulling a polarization bound back through $q$ costs a factor $(1+\varepsilon)^k$, whose $k$-th root tends to $1$, completing the proof.

Load-bearing premise

The proof leans on an exact formula for the polarization constants of finite-dimensional complex $\ell_1$ spaces, taken as given; if that formula were false, the proof that these constants have $k$-th root tending to $1$ would collapse.

Editorial extensions

If this is right

  • For holomorphic functions on a finite-dimensional complex space, the radius of convergence equals the radius computed from the symmetric multilinear Taylor coefficients: $R_a(f)=R_a^{\mathrm{mult}}(f)$ (Corollary 2.4).
  • A monomial power series converges absolutely and uniformly on the full polydisc of radius $R$, rather than only on $(1/e)R$ of that polydisc, when the space is finite-dimensional and complex (Proposition 2.5).
  • In every finite-dimensional real space, the polarization constant is at most $2$ and equals the complexification constant, so the finite-dimensional complex theorem marks a sharp real/complex divide.
  • For an infinite-dimensional complex space, having $\mathbf{c}(2,X)=1$ forces type $2$ and cotype $2$, placing $X$ in the same isomorphic class as inner-product spaces (Theorem 3.1).
  • The best constant for multiplying nuclear polynomials satisfies $m(k_1,\ldots,k_n,X)=\mathbf{c}(k_1,\ldots,k_n,X^*)$, transferring all polarization results to nuclear-norm product estimates (Theorem 4.1).

Reading between the lines

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

  • One testable extension is to replace $\ell^d_1(\mathbb{C})$ by $\ell^d_p(\mathbb{C})$ for $1<p<\infty$; with an exact or asymptotic formula there, the same quotient technique could give explicit convergence rates for $\mathbf{c}(k,X)^{1/k}$, which the paper does not quantify.
  • The theorem makes the asymptotic polarization constant a trivial invariant for finite-dimensional complex spaces, so finer distinctions must come from finite-degree constants $\mathbf{c}(k,X)$ or from the real case; classifying spaces with the symmetric operator norm property remains an open geometric question.
  • Because equality of the two radii is equivalent to $\mathbf{c}(X)=1$, the paper implies that on finite-dimensional complex domains, Taylor-series convergence is governed by the symmetric multilinear forms alone, which may simplify how convergence radii are computed in several complex variables.
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Editorial analysis

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

Referee Report

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Summary. The paper proves that the polarization constant c(X) = limsup_{k→∞} c(k,X)^{1/k} of every finite-dimensional complex Banach space X is equal to 1 (Theorem 1.1). The proof first establishes c(ℓ_1^d(C)) = 1 using Sarantopoulos's closed formula (9) for c(k, ℓ_1^d(C)) and an exact balancing argument. It then shows, via an almost-isometric quotient lemma (Lemma 2.2), that every finite-dimensional complex space is an almost-isometric quotient of ℓ_1^d(C), so the polarization constants of X are bounded above by those of ℓ_1^d(C) up to a factor (1+ε)^k. Letting ε→0 yields c(X) ≤ 1, while c(X) ≥ 1 is trivial. The remainder of the paper develops consequences: equality of the two notions of radius of convergence for holomorphic functions (Corollary 2.4), the failure of the theorem over the reals together with c(X)=b(X)≤2 for real finite-dimensional spaces (Proposition 2.7), type/cotype restrictions for spaces with the symmetric operator norm property (Theorem 3.1), and identities relating polarization constants to nuclear norms of products of polynomials (Theorem 4.1 and subsequent corollaries).

Significance. If correct, Theorem 1.1 is a striking and natural result: every finite-dimensional complex Banach space has the same asymptotic polarization constant as Hilbert space, saturating the trivial lower bound 1. The proof is elegant and, apart from the black-box use of Sarantopoulos's formula, essentially self-contained; the quotient-lifting argument is a nice technique. The paper also contains several auxiliary results of independent interest: the identification of the polarization constant with the Bochnak constant in the real setting, the type/cotype necessary conditions, and the sharp identities for nuclear norms of products of polynomials. The exposition is clear overall, and the main theorem is supported by a sound chain of reasoning with no adjustable parameters.

minor comments (5)
  1. [§2, proof of Theorem 1.1] The sentence immediately before inequality (12) says that the multilinear form ∨P∘(q,...,q) has norm less than or equal to one; this is not true in general (its norm can be as large as c(k, ℓ_1^d(C))‖P‖). The displayed inequality (12) uses the correct bound via ‖P∘q‖ ≤ ‖P‖, so the argument is valid, but the offending sentence should be corrected or removed.
  2. [§2.1, proof of Proposition 2.3] In the construction of the counterexample, the series f = ∑_{j} P_{k_j} has radius of convergence 1 and is not defined on all of X; the text says “f ∈ H(X)”, which should be “f ∈ H(B_X)” or “f defined in a neighborhood of 0”. The contradiction argument still works with this clarification.
  3. [§2.1, equation (18) and surrounding text] The chain of inequalities leading to R(f)/(c(X)+ε) ≤ R_mult(f) ≤ R(f) is correct, but the presentation would be clearer if the definition of radius of convergence via limsup were invoked explicitly, since the notation R and R_mult is used without restating formulas (14)–(15).
  4. [§3, proof of Lemma 3.4] In the lower-bound computation for ‖∨P‖, the sum of the three moduli equals 6 because λ+λ^2 = −1 and |λ−λ^2| = √3; spelling this out would improve readability and avoid leaving the reader to verify the value 6.
  5. [Throughout] There are numerous typographical and OCR-style errors, including “Bellow” for “Below”, “concuide” for “conclude”, “T ake” for “Take”, “consid er” for “consider”, and inconsistent spacing in displayed formulas. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem follows from an external exact formula for ℓ_1^d and a quotient argument; nothing is fitted, renamed, or reduced to its own input.

full rationale

The derivation chain is self-contained modulo ordinary external citations. Theorem 1.1 is proved by reducing an arbitrary finite-dimensional complex space X to a finite-dimensional ℓ_1^d space: Lemma 2.2 provides a norm-one quotient q:ℓ_1^d→X with almost-isometric lifting, giving c(k,X) ≤ (1+ε)^k c(k,ℓ_1^d(C)) in inequality (11). Proposition 2.1 then evaluates c(ℓ_1^d(C))=1 using Sarantopoulos's exact formula (9), quoted from [26, Proposition 4] as an independent published result: c(k,ℓ_1^d(C)) = max{ k_1!⋯k_d!/k! · k^k/(k_1^{k_1}⋯k_d^{k_d}) }. The proof of Proposition 2.1 only analyzes the maximizer (10) and takes the k-th root limit; no parameter is adjusted to force the value 1, and the lower bound c(X)≥1 is the trivial polarization bound (2). The cited formula is genuine external evidence: it gives exact values for every k, is parameter-free, is not derived from Theorem 1.1, and its d=2,k=2 specialization reproduces the known value 2. The later real-case Proposition 2.7 uses Theorem 1.1 only for the complexification of a real space, which is an application rather than an input. Self-citations in the bibliography are not load-bearing: the cited [9] concerns linear polarization constants and is not used in the main proof. No equation in the paper reduces to its own target by construction, and no fitted quantity is renamed as a prediction. Hence there is no circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorem has no free parameters and no invented entities. It relies on external classical theorems (Sarantopoulos's formula, Banach's Hilbert-space result, Bochnak complexification bounds, interpolation theory) that are standard in the field. The type/cotype duality facts are used without proof but are routine. No assumption is ad hoc to this paper.

assumptions (6)
  • domain assumption Sarantopoulos's exact formula (9): c(k, ℓ^d_1(C)) = max{ k_1!...k_d! / k! * k^k / (k_1^{k_1}...k_d^{k_d}) }.
    Used as the starting point for Proposition 2.1. It is cited from [26, Proposition 4] and not reproved in the paper.
  • standard math Banach's theorem: for Hilbert spaces H, ||∨P|| = ||P|| for every k-homogeneous polynomial P.
    Cited in the introduction and used in Section 3 for the Hilbert case c(2,H)=1.
  • domain assumption The Bochnak complexification constant satisfies b(k,X) ≤ 2^{k-1} for finite-dimensional real spaces ([19, Proposition 18]).
    Used in Proposition 2.7 to bound the real polarization constant by 2.
  • standard math Complex interpolation of polynomials: for compatible Banach couples, the interpolated polynomial norm is at most M_0^{1-θ} M_1^θ.
    Used in Lemma 3.4; the paper proves a version as Proposition 3.3 using the external interpolation theorem [8, Theorem 4.4.1].
  • domain assumption Quotient lower bound for polarization constants ([25, Lemma 0]): c(2,X) ≥ c(2, X/Y)/(1+ε)^2 for (1+ε)-isomorphic quotients.
    Used in the proof of Theorem 3.1 to transfer lower bounds from ℓ^d_p to X.
  • standard math Standard duality relations between type and cotype: p(X**) = q(X*)' and q(X**) ≤ p(X*)'.
    Used in the last step of the proof of Theorem 3.1 without proof.

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Pith. "Pith review of The polarization constant of finite dimensional complex spaces is one." pith.science (2026). https://pith.science/paper/Z5PV4OQO

@misc{pith2026190808107,
  author       = {Pith},
  title        = {Pith review of: The polarization constant of finite dimensional complex spaces is one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5PV4OQO}},
  note         = {Machine review of arXiv:1908.08107}
}
abstract

The polarization constant of a Banach space $X$ is defined as $$\mathbf c(X):= \limsup\limits_{k\rightarrow \infty} \mathbf c(k, X)^\frac{1}{k},$$ where $\mathbf c(k, X)$ stands for the best constant $C>0$ such that $ \Vert \overset{\vee}{P} \Vert \leq C \Vert P \Vert$ for every $k$-homogeneous polynomial $P \in \mathcal P(^kX)$. We show that if $X$ is a finite dimensional complex space then $\mathbf c(X)=1$. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak's complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related with the geometry of $X$ for $\mathbf c(2,X)=1$ to hold. Additionally we link polarization's constants with certain estimates of the nuclear norm of the product of polynomials.

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