Strictly Stable Hurwitz Polynomials and their Determinantal Representations

Hugo J. Woerdeman; Victor Vinnikov

REVIEW 3 major objections 5 minor 46 references

Strictly Stable Hurwitz Polynomials and their Determinantal Representations

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Every strictly stable Hurwitz polynomial admits, up to a stable factor, a certifying determinantal representation as the determinant of a linear pencil with positive definite imaginary part.

desk verdict Theorem 3.1 is sound and the paper is a real step toward certifying representations on tube domains, but the abstract oversells the conjectural Section 8 and a few proofs need cleanup before publication. read the letter →

arxiv 2411.17526 v1 pith:DBLNGQ4G submitted 2024-11-26 math.FA math.AG

Victor Vinnikov , Hugo J. Woerdeman This is my paper · ORCID

classification math.FAmath.AG MSC 15A1526C1032A1047A1330C1093B28

keywords DeterminantalrepresentationHurwitzstabilitySchurMultivariablepolynomialSiegeldomainLieballTubeLorentzcone

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

This paper proves a multivariable Hurwitz stability version of the known Schur-stability determinantal representation theorem. If a polynomial $p$ of multidegree $(n_1,\dots,n_d)$ satisfies the growth bound $|p(z)| \ge \epsilon \prod_{j=1}^d |z_j + i|^{n_j}$ on the upper half plane $\mathbb{H}^d$—the paper's notion of strict stability—then there is a polynomial $q$ for which $pq$ admits a certifying determinantal representation $pq = \det(A_0 + z_1 A_1 + \cdots + z_d A_d)$ with $\operatorname{Im} A_0 > 0$, $A_j \ge 0$, and $A_1 + \cdots + A_d > 0$. The argument runs through the Cayley transform from the unit disk to the upper half plane, so the added factor is what makes the product fit into the Schur-side contractive representation. The same pattern is extended to tube domains over matrix and Siegel upper halfspaces, skew-symmetric matrix halfspaces, and Lorentz cones; for the exceptional 27-variable tube domain the result is established conditional on a stated polynomial-convexity conjecture. The reason to care is that certifying determinantal representations of this kind exhibit stability through matrix positivity, which is the bridge to semidefinite programming and to the generalized Lax conjecture for hyperbolic polynomials.

What carries the argument

The machinery is the Cayley transform $\varphi(z) = i(1+z)/(1-z)$, which maps the unit disk conformally onto the upper half plane and converts Hurwitz stability into Schur stability. On the unit-disk side, the paper invokes the Matrix-valued Hermitian Positivstellensatz (specifically [11, Theorem 4.1]), which gives a contractive determinantal representation $\tilde p \tilde q = \tilde p(0)\tilde q(0) \det(I - K Z_N)$ for strictly stable Schur polynomials. Pulling that representation back through the Cayley transform yields $A_0 = i(I+K)(I-K)^{-1}$, which has positive definite imaginary part, and the block projections $A_j$ that are positive semidefinite and sum to the identity.

What would settle it

Construct a polynomial of multidegree $(n_1,\dots,n_d)$ that satisfies the growth bound (3.6) but for which no polynomial $q$ makes $pq$ equal to $\det(A_0 + z_1 A_1 + \cdots + z_d A_d)$ with $\operatorname{Im} A_0 > 0$ and all $A_j \ge 0$; such an example would refute Theorem 3.1. A smaller, still decisive test is to check computationally, for low degrees and $d=2$, whether every Schur-side representation from [11, Theorem 4.1] pulled back through the Cayley transform can be chosen so that denominator factors cancel to a polynomial $q$.

Watch

Extended reading notes

Core claim

Working in $\mathbb{H}^d = \{z \in \mathbb{C}^d : \operatorname{Im} z_j > 0\}$, the paper's central theorem (Theorem 3.1) asserts the following. Let $p$ have multidegree $(n_1,\dots,n_d)$ and suppose there is $\epsilon > 0$ with $|p(z)| \ge \epsilon \prod_{j=1}^d |z_j + i|^{n_j}$ for all $z \in \mathbb{H}^d$. Then there exists a polynomial $q$ and matrices $A_0,\dots,A_d$ such that $\operatorname{Im} A_0 > 0$, $A_j \ge 0$, $A_1 + \cdots + A_d > 0$, and $p(z)q(z) = \det(A_0 + z_1 A_1 + \cdots + z_d A_d)$. The proof Cayley-transforms the upper half plane to the unit disk, applies the Schur-stable contractive determinantal representation, and transforms back; the growth bound is exactly what makes the transformed polynomial bounded away from zero on the closed polydisk.

Load-bearing premise

Everything rests on the growth lower bound $|p(z)| \ge \epsilon \prod_{j=1}^d |z_j + i|^{n_j}$ holding on the whole upper half plane; if a stable polynomial decays faster near infinity, the Cayley transform no longer lands in the setting where the known Schur-side representation applies.

Editorial extensions

If this is right

Every polynomial satisfying the growth bound (3.6) admits, after multiplication by a certifying factor $q$, a determinantal representation $\det(A_0 + \sum_j z_j A_j)$ with $\operatorname{Im} A_0 > 0$ and each $A_j \ge 0$.
Proposition 3.2 characterizes the growth bound: it is equivalent to the Cayley-transformed polynomial being bounded away from zero on the closed polydisk, and also to $p$ containing the full multidegree term $\prod_j z_j^{n_j}$ while staying bounded away from zero on $\mathbb{H}^d$.
The same method gives certifying determinantal representations for polynomials strictly stable on tube domains over matrix and Siegel upper halfspaces, skew-symmetric matrix halfspaces, and the bivariable and $n$-variable Lorentz cones.
For the exceptional 27-variable tube domain, the analogous representation follows if a stated polynomial-convexity conjecture about the bounded domain $C$ is resolved.
The result provides an upper-half-plane analogue of the known unit-disk contractive determinantal representation theorem, placing strict Hurwitz stability within the same certifying-representation framework as Schur stability.

Reading between the lines

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

Editorial extension: if Theorem 3.1 is correct, the growth bound (3.6) is the natural notion of strictness in multivariable Hurwitz stability, and the same bound would be the first hypothesis to try in any search for a constructive version of the generalized Lax conjecture beyond the plane-curve case.
Editorial extension: because the proof reduces to a Schur-side statement, an effective algorithmic version could be obtained by tracking the matrix sizes in [11, Theorem 4.1]; numerical experiments on low-degree examples could reveal how large the certifying factor $q$ must be.
Editorial extension: in the exceptional 27-variable case, resolving polynomial convexity of $C$ is the only missing ingredient; the paper's own suggested route—proving rotational invariance of $C$—would likely complete the theorem by falling into the known theory of bounded symmetric domains.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the main theorem is fine; the packaging is not. Section 3's Theorem 3.1 does what it claims: under the growth bound (3.6), a Cayley transform turns p into a strongly stable Schur polynomial, [11, Thm 4.1] supplies a contractive determinantal representation, and transforming back gives Im A0 > 0 with the A_j nonnegative projections. I checked the algebra in (3.7) and it is correct. The strict stability hypothesis is strong—it forces the top multidegree term and lower bounds on |p|—but that is a scope limit, not a flaw, and Proposition 3.2 says exactly what it buys you.

What is genuinely new is carrying this template from the polydisk to tube domains over symmetric cones. The sections on matrix, skew-symmetric, and Lorentz cones follow the same Cayley/reduction pattern, and the n-variable Lorentz section introduces a genuine matrix-pencil description of the Lie ball, P±(z), which is a real contribution and likely to be reusable.

Now the soft spots, in proportion. The abstract says "we establish various certifying determinantal representation results" but Section 8 is explicitly conjectural. That should be in the abstract. The proof of Theorem 7.1 refers to "Conjecture 7.7" when the proved Theorem 7.7 is meant; that is just a typo, but confusing because there is also a real Conjecture 8.7. Theorem 7.7 itself is presented as a variation of [11, Theorem 4.1] and is only sketched; a referee will need to check that the polynomial convexity argument and the operator-tuple maximalization really feed in as claimed. Lemma 6.2 contains a wrong matrix computation: the 2x2 matrix displayed does not have zero trace, and its eigenvalues are not the two nonzero eigenvalues of zz* − zz^T. The final statement of Lemma 6.2 is true, so this is a fixable proof error, not a counterexample.

Is the reliance on [11] a problem? No. The cited theorem is peer-reviewed and the new content is the transport theorem and the cone-by-cone analysis. The paper is not self-contained, though; readers need [11] in hand.

Who should read this: people working on the generalized Lax conjecture, stable polynomials, and determinantal representations. It is not a breakthrough that resolves Lax, but it is a meaningful step and the Lie ball pencil is worth having. I would send it to a serious referee. My own verdict would be conditional: accept after fixing the abstract, the reference typo, and the proof of Lemma 6.2, and after a specialist checks the sketch of Theorem 7.7.

Referee Report

3 major / 5 minor

Summary. The paper proves a family of certifying determinantal representation theorems for multivariable polynomials that are strictly stable on tube domains. The main result (Theorem 3.1) shows that if a polynomial p of multidegree (n_1,...,n_d) satisfies the growth lower bound |p(z)| ≥ ε ∏_j |z_j + i|^{n_j} on the upper half plane H^d, then there exists a polynomial q such that pq equals det(A_0 + z_1 A_1 + ... + z_d A_d) with Im A_0 > 0, A_j ≥ 0, and A_1 + ... + A_d > 0. The proof uses the Cayley transform to reduce to a strictly Schur-stable polynomial on the polydisk and invokes the matrix-valued Hermitian Positivstellensatz of [11] (Grinshpan et al.). Sections 4–7 extend this approach to tube domains over matrix, Siegel, skew-symmetric, and Lorentz cones, and Section 8 formulates conjectural results for the exceptional 27-dimensional tube domain.

Significance. Theorem 3.1 is a clean and largely convincing application of [11, Theorem 4.1] and provides a new class of strictly stable Hurwitz polynomials with certifying determinantal representations, connecting naturally to the generalized Lax conjecture. The algebraic computations in the proof of Theorem 3.1 are explicit and checkable, and the Cayley-transform strategy is elegant. The extension to symmetric cones is potentially significant, but the proof of the n-variable Lorentz cone case (Theorem 7.7) is only sketched, so the full scope of the paper's claims is not yet established. If the missing details are supplied, the paper would make a solid contribution to the operator-theoretic approach to stable polynomials and determinantal representations.

major comments (3)

[Theorem 7.7] The proof of Theorem 7.7 is not complete. The statement 'We will apply a variation of [11, Theorem 4.1] where the inequality I - P(z)^*P(z) > 0 is replaced by P_+(z)^*P_+(z) - P_-(z)^*P_-(z) > 0' is a substantial generalization, and the subsequent sentence 'Adjusting now the proof of [11, Theorem 4.1] yields the desired determinantal representation' does not show how the lurking-contraction argument is adapted. In particular, the application of [11, Theorem 2.3] to the noncommutative polynomial P_+^*P_+ - P_-^*P_- is not demonstrated. Since Theorem 7.7 is used to prove Theorem 7.1, this gap affects the Lorentz-cone results and the conjectural Section 8. The authors should provide a full proof or a precise statement of the theorem from [11] that covers this setting.
[Sections 4–6] The paper repeatedly applies [11, Theorem 4.1] to domains that are not the polydisk, such as products of Cartan type I and III domains (Theorem 4.1), the skew-symmetric domain (Theorem 5.2), and the 2-variable Lie ball (Theorem 6.3). The introduction only states the polydisk version of [11, Theorem 4.1]. For the reader to verify these applications, the authors should either state the full matrix-valued version of [11, Theorem 4.1] they are using, or explain why the polydisk version suffices after a Cayley transform. This is particularly relevant in Theorem 6.3, where the Lie ball is described by I - P(z)^*P(z) > 0 with a 2×2 matrix P(z), and the claim that [11, Theorem 4.1] applies is not immediate.
[Theorem 7.1, proof] The proof of Theorem 7.1 contains the sentence 'By the assumption that Conjecture 7.7 holds...' but there is no Conjecture 7.7; the reference should be to Theorem 7.7. While this is a typo, it appears in a load-bearing step and, as written, suggests an unproved assumption. The authors should correct this and ensure that all references to numbered results are accurate.

minor comments (5)

[Lemma 6.2] The displayed formula for M(z) contains the term zz^* - zz^T with off-diagonal entries printed as z1 z2 - z1 z2, which is identically zero; the intended matrix appears to be M(z) = ‖z‖^2 I_n - zz^* + zz^T. Please fix the display.
[Theorem 3.1, proof] After equation (3.7), the definition 'A_j = ⊕_{k=1}^d δ_{jk} I_{N_k}, j = 1,...,k' should read j = 1,...,d; also, the phrase 'increase N_j in (7)' should refer to equation (3.7).
[Theorem 3.1, proof] The choice of r>1 for which ilde p(rz) is strictly Schur stable on ar D^d is not explained; since ilde p is zero-free on the closed polydisk, one can take r>1 sufficiently close to 1, but this should be stated explicitly.
[Proposition 3.2] In the proof of (iii) ⇒ (i), the dominance argument for the full multidegree term is stated only for |z_1|+...+|z_d| ≥ M; the case where some variables stay bounded while others tend to infinity should be addressed explicitly.
[Throughout] There are numerous typographical errors (e.g., 'coeeﬁcients' in the introduction, 'Lorents cone' in the Section 6 heading, 'peroperty' in the footnote on page 23, and the reference to 'Conjecture 7.7' already noted). A careful proofreading is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: Theorem 3.1 is a genuine Cayley reduction to the independent published theorem [11, Thm. 4.1]; the partial author overlap does not make the argument circular.

full rationale

The paper's central claim, Theorem 3.1, is not defined in terms of its conclusion. The strict-stability hypothesis (3.6) is a growth lower bound on H^d, and Proposition 3.2 derives equivalent conditions internally (full multidegree term and a uniform lower bound); none of these conditions mentions determinantal representations. The proof Cayley-transforms p to a polynomial p-tilde on the polydisk and verifies from (3.6) that |p-tilde| >= eps * 2^{sum n_j}, so p-tilde is strongly Schur-stable. At that point the paper invokes [11, Theorem 4.1], which supplies a strictly contractive K and a polynomial q-tilde with p-tilde*q-tilde = det(I - K Z_N). The reverse Cayley computation (3.7) is explicit: A0 = i(I+K)(I-K)^{-1} has Im A0 > 0, the A_j are nonnegative projections summing to the identity, and no commutativity with the diagonal is silently assumed. The potential denominator issue is addressed by enlarging N_j and adding zero rows and columns to K, which is a legitimate bookkeeping step. The heavy use of [11] is a self-citation in the sense that Vinnikov and Woerdeman are among the authors of [11], but [11] is a published, parameter-free theorem whose assumptions (Schur stability on the polydisk) are distinct from the paper's tube-domain conclusions; it is therefore independent support rather than a circular input. The later sections (4-7) repeat the same honest reduction for matrix, skew-symmetric, and Lorentz-cone tubes, and Section 8 explicitly identifies the missing ingredient (Conjecture 8.7, polynomial convexity) rather than smuggling it in. The typographical issues noted by the reader, such as the garbled matrix display in Lemma 6.2 and the reference to 'Conjecture 7.7' in the proof of Theorem 7.1, are editorial and do not affect the derivation chain. Overall, no step reduces by construction to its own input; the modest score reflects only the prominence of the partially overlapping prior work, not actual circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard operator-theoretic and real-algebraic background, chiefly the self-cited contractive determinantal representation theorem of [11], polynomial convexity of the Lie ball, von Neumann's inequality, and Cayley transform bijections for symmetric cones. The results introduce no free parameters and no new physical or algebraic entities. A local proof error in Lemma 6.2 and the 'Conjecture 7.7' typo are documented in red_flags; they do not by themselves change the ledger.

assumptions (5)

standard math Contractive determinantal representation for strictly stable Schur polynomials ([11, Theorem 4.1])
Central engine: the present paper reduces tube-domain problems to this theorem via the Cayley transform. Used in Theorems 3.1, 4.1, 5.2, 6.3, 7.7. The theorem is peer-reviewed but authors overlap with the present paper.
standard math Polynomial convexity of the Lie ball L_n (from convexity, see [25])
Needed in Theorem 7.7 to prove boundedness in inequality (7.17) and to conclude the variant of [11, Theorem 4.1] used for the Lie ball.
standard math Cayley transform bijections between bounded symmetric domains and tube domains (e.g., Proposition 7.2 for L_n to TC_n)
Bridges the Schur-stable setting of [11] to the Hurwitz/tube-domain setting; formulas are stated and partly proved in the text, relying on standard results like [6, Theorem X.4.3].
standard math Von Neumann's inequality and spectral mapping for rational functions of commuting contractions
Used in Lemma 7.5 to show that a commuting tuple satisfying the P± matrix inequality has joint spectrum in the Lie ball.
standard math Garding's theory of hyperbolic polynomials, including interlacing of roots
Foundation of Section 2, connecting Omega_C-stability to hyperbolicity of the leading homogeneous part; referenced to [7,8,22].

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Pith. "Pith review of Strictly Stable Hurwitz Polynomials and their Determinantal Representations." pith.science (2026). https://pith.science/paper/DBLNGQ4G

@misc{pith2026241117526,
  author       = {Pith},
  title        = {Pith review of: Strictly Stable Hurwitz Polynomials and their Determinantal Representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DBLNGQ4G}},
  note         = {Machine review of arXiv:2411.17526}
}

read the original abstract

We establish various certifying determinantal representation results for a polynomial that contains as a factor a prescribed multivariable polynomials that is strictly stable on a tube domain. The proofs use a Cayley transform in combination with the Matrix-valued Hermitian Positivstellensatz developed in ArXiv:1501.05527.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

Grinshpan, D.S

A. Grinshpan, D.S. Kaliuzhnyi-Verbovetskyi, V. Vinni kov, and H.J. Woerde- man. Matrix-valued Hermitian Positivstellensatz, lurkin g contractions, and con- tractive determinantal representations of stable polynom ials. Oper. Th. Adv. Appl., 255 (2016), 123–136

work page 2016

[2] [1]

Atiyah, R

M. Atiyah, R. Bott, and L. G ˚ arding. Lacunas for hyperbol ic diﬀerential opera- tors with constant coeﬃcients, I. Acta Math. , 124:109–189, 1970

work page 1970

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