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Any-dimensional Positivstellens\"atze for symmetric functions

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Pith's one-line read Symmetric functions bounded below by a fixed ε > 0 on every unit sphere admit dimension-independent Pólya and Reznick certificates via powers of the second power sum.

desk verdict Solid any-dimensional Pólya/Reznick certificates for symmetric functions, with a clean orbit-space/moment characterization; the long measure classification is the only intricate step and it holds up. read the letter →

arxiv 2606.27551 v2 pith:I6C2BMMJ submitted 2026-06-25 math.AG math.FAmath.OC

Sebastian Debus , Robin Schabert This is my paper
classification math.AGmath.FAmath.OC MSC 14P1013J3005E05
keywords Positivstellensatzsymmetricfunctionssumsofsquarespowerany-dimensionalmomentmeasuresinfinitegroupPólyatheorem
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 two uniform positivity certificates for symmetric functions that stay strictly positive by a fixed amount in every dimension. If a homogeneous even symmetric function is at least ε > 0 on every sphere, some power of the second power sum multiplies it into a non-negative linear combination of even monomial symmetric functions. For homogeneous functions written in power sums starting from p₂ that are likewise bounded below by ε on the unit sphere, a high enough power of p₂ turns the product into a sum of squares that works simultaneously for every number of variables. The argument identifies the dual of a natural Archimedean preordering with moment sequences of discrete probability measures on [−1,1] arising from unit vectors in ℓ²; that description also realises the closed orbit space of the infinite symmetric group. An elementary reduction recovers the known certificates for the normalised (power-mean) setting from classical real-algebraic geometry on Hankel spectrahedra.

What carries the argument

The Archimedean preordering T = (any-dimensional sums of squares inside A) + (1 − p₂) in the ring A = ℝ[p₂, p₃, …], whose character space is precisely the product-topology closure of the sequences (p₂(x), p₃(x), …) for unit vectors x ∈ ℓ²; the representation theorem then forces the function into T, after which a substitution and clearing of denominators yields the certificate.

What would settle it

Exhibit a homogeneous f in the power sums from p_{2} onward that stays at least ε > 0 on every unit sphere, yet for every k the product p_{2}^k f fails to be a sum of squares after truncation to some large finite number of variables.

Watch

Extended reading notes

Core claim

Two any-dimensional Positivstellensätze hold under a uniform gap: Corollary 3.3 (Pólya type) produces non-negative coefficients in the even monomial basis after multiplication by p₂^k, and Theorem 3.7 (Reznick type) produces an any-dimensional sum of squares after multiplication by p₂^k, for homogeneous functions free of p₁ that satisfy f ≥ ε > 0 on every unit sphere.

Load-bearing premise

Every ring homomorphism non-negative on the preordering must arise as a limit of discrete probability measures coming from unit vectors in ℓ²; if an exotic character were missed, the representation theorem would not force the certificate.

Editorial extensions

If this is right

  • Uniform lower bounds become decidable for even homogeneous symmetric functions via the explicit Pólya certificate.
  • Mere non-negativity is insufficient: there exist any-dimensional non-negative functions with bad points for which no power of p_{2} yields an any-dimensional sum of squares.
  • The closed orbit space of the infinite symmetric group acting on the unit sphere in ℓ² is identified with a concrete set of discrete moment sequences on [−1,1].
  • Normalised symmetric functions reduce directly to classical Positivstellensätze on Hankel spectrahedra, recovering Krivine–Stengle-type certificates.

Reading between the lines

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

  • The same discrete-measure dictionary may supply certificates for other representation-stable positivity problems beyond pure symmetry.
  • Whether the uniform gap ε > 0 can be relaxed to mere pointwise positivity (infimum zero) remains open and would require controlling escaping sequences of measures.
  • Effective bounds on the exponent k, analogous to classical Powers–Reznick estimates, are not supplied and could be tested numerically on low-degree examples.
  • Dropping evenness or the uniform gap may render any-dimensional non-negativity undecidable, consistent with known multi-homogeneous hardness results.
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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

0 major / 5 minor

Summary. The paper establishes two any-dimensional Positivstellensätze for symmetric functions that are uniformly bounded below by a positive constant ε. Corollary 3.3 is a Pólya-type result: a homogeneous even symmetric function f with f^(n) ≥ ε on every unit sphere admits a power of p₂ that expands with non-negative coefficients in the even monomial-symmetric basis. Theorem 3.7 is a Reznick-type result: a homogeneous f ∈ ℝ[p₂,…,p_{2d}] with f ≥ ε on the unit sphere of ℓ₂ becomes any-dimensional sos after multiplication by a sufficiently high power of p₂. The proof of the latter proceeds by showing that the preordering T = (SΣ ∩ A) + (1-p₂) in A = ℝ[p₂,p₃,…] is Archimedean (Lemma 3.10), classifying its character space K_T as the product-topology closure of the moment sequences of discrete probability measures on [-1,1] arising from unit vectors in ℓ₂ (Lemma 4.6 / Corollary 4.7), and applying the Jacobi–Marshall representation theorem. The same classification yields a description of the S_∞-orbit space of the unit sphere in ℓ₂. Section 5 constructs homogeneous examples with bad points showing that the strict lower bound ε cannot be relaxed to mere non-negativity. Section 6 supplies an elementary alternative proof of known Positivstellensätze for normalized symmetric functions via Hankel spectrahedra and classical Krivine–Stengle/Schmüdgen/Putinar theorems.

Significance. The results give the first algebraic positivity certificates that are uniform across all dimensions for ordinary (non-normalized) symmetric functions, a setting whose natural domain is not semialgebraic. The moment-theoretic identification of K_T and the orbit-space description (Corollary 4.7) are of independent interest and cleanly link real algebra, classical moment problems, and representation theory of S_∞. The necessity of the ε-hypothesis is demonstrated by explicit bad-point constructions (Example 5.6), and the normalized case is recovered by a short, self-contained argument that clarifies the relation to existing work on pure moment and trace polynomials. The proofs are complete, rely only on standard tools (Markov–Lukács, Riesz–Haviland, Weierstrass approximation, Jacobi–Marshall), and do not introduce free parameters or ad-hoc axioms.

minor comments (5)
  1. In the abstract and introduction the phrase “infinite dimensional analogous” should be “infinite-dimensional analogues” (or “analogs”).
  2. Lemma 4.6 is long and dense; a short roadmap paragraph at the beginning of the lemma (listing the six claims and the tools used for each) would improve readability without changing the mathematics.
  3. In the proof of Theorem 3.7 the substitution X_i / √p₂ is performed in the quotient field of formal power series; a one-sentence remark that the resulting identity is first obtained formally and then cleared of denominators would make the passage fully explicit.
  4. Example 5.6 writes the power-sum expression for A_m; verifying the coefficients by a short computer-algebra check (or citing a notebook) would strengthen reproducibility.
  5. A few typographical slips remain: “any-dimensional sos” is sometimes written without the hyphen, and the arXiv identifier in the header is dated 2026.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; main certificates follow from independent Archimedean representation plus moment classification of characters, with only non-load-bearing self-cites for context/examples.

  1. self citation load bearing [Theorem 2.3 and surrounding discussion in §2; Examples 2.5, 5.6]
    "Theorem 2.3([2] Theorem 3.12).For any even degree2d≥4the relationSΣ =2d ⊊SP=2d holds. Moreover, in all these cases the setSΣ=2d is semialgebraic butSP=2d is not semialgebraic. ... A homogeneous example of an element inSP∖SΣis f(n)=4(p(n)1)4−5p(n)2(p(n)1)2−13920p(n)3p(n)1+4(p(n)2)2+4p(n)4 which is not sos forn≥4(see[2, Theorem 3.6])."

    Citation [2] has overlapping authorship (Debus). It supplies the qualitative distinction SΣ⊊SP and concrete examples used for motivation and to show ε>0 is necessary (via bad points). However the citation is not load-bearing for the certificates themselves: the proofs of Corollary 3.3 and Theorem 3.7 never invoke the non-semialgebraicity or the specific examples as premises; they rely only on the independent Archimedean representation and the moment classification of KT. Thus the self-cite is present but non-circular for the main claims.

full rationale

The derivation chain for the central claims (Corollary 3.3 via Powers-Reznick bound independent of n; Theorem 3.7 via Jacobi-Marshall representation on the Archimedean preordering T=(SΣ∩A)+(1-p2) after classifying KT as the product-topology closure of unit-sphere moments of discrete measures on [-1,1]) is self-contained. Lemma 3.10 proves Archimedeanness by explicit univariate sos lifts of t2±tk; Lemma 4.6 proves the character classification by Riesz-Haviland + Markov-Lukács (moments), Weierstrass + e2-sos (Lemma 4.4) (discreteness/isolation), and falling-factorial evaluation of ek (Lemma 4.5) (integer multiplicities), none of which presuppose the Positivstellensatz conclusion. The subsequent substitution Xi/√p2 and clearing denominators is pure algebra. Self-citations to [2] (overlapping authors) appear only for qualitative SΣ vs SP comparisons, degree-2 characterizations, and examples of elements in SP∖SΣ; they are not used as premises for the certificates or the KT classification. Normalized results in §6 reduce directly to classical Krivine-Stengle/Schmüdgen/Putinar on Hankel spectrahedra via the structural observation of [1] and the elementary Gram factorization of principal minors (Lemma 6.4). No fitted parameters, self-definitional loops, uniqueness imports, or ansatz smuggling occur. Score 1 only for the presence of non-load-bearing self-cites.

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

The paper works entirely inside classical real algebra and moment theory. No numerical parameters are fitted; the only background results invoked are standard theorems (Jacobi–Marshall representation, Markov–Lukács, Riesz–Haviland, Weierstrass approximation, Newton identities) whose statements are used as black boxes. The preordering T and the ring A are ordinary constructions, not new physical or mathematical entities.

assumptions (4)
  • standard math Jacobi–Marshall representation theorem for Archimedean T-modules (Thm. 3.8 / [21, 5.4.4])
    Applied directly to conclude f∈T once positivity on K_T and Archimedeanness are established.
  • standard math Markov–Lukács theorem: non-negative polynomials on [−1,1] admit sos representations of the form σ1+(1−t²)σ2
    Used to prove Archimedeanness of T (Lemma 3.10) and to construct test elements in the character-space analysis.
  • standard math Riesz–Haviland theorem characterizing moment sequences of measures on compact intervals
    Invoked in Lemma 4.6(1) to relate non-negative linear functionals to measures.
  • domain assumption The ring of symmetric functions Λ=ℝ[p1,p2,…] and its truncations behave as stated under evaluation on ℓp and ℝ∞
    Standard combinatorial algebra; used throughout to identify any-dimensional positivity with positivity on ℓ1/ℓ2.

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Pith. "Pith review of Any-dimensional Positivstellens\"atze for symmetric functions." pith.science (2026). https://pith.science/paper/I6C2BMMJ

@misc{pith2026260627551,
  author       = {Pith},
  title        = {Pith review of: Any-dimensional Positivstellens\"atze for symmetric functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6C2BMMJ}},
  note         = {Machine review of arXiv:2606.27551}
}
abstract

Positivstellens\"atze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellens\"atze for symmetric functions that are uniformly bounded below by some $\varepsilon > 0$. These are infinite-dimensional analogs of theorems of P\'olya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots)$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the closure of the orbit space of the infinite symmetric group on the sphere. Finally, we provide an alternative proof of existing Positivstellens\"atze for normalized symmetric functions.

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