Minimal polynomials, scaled Jordan frames, and Schur-type majorization in hyperbolic systems

Juyoung Jeong; M. Seetharama Gowda; Sudheer Shukla

arxiv: 2603.10522 · v2 · pith:BDX4PXJNnew · submitted 2026-03-11 · 🧮 math.OC

Minimal polynomials, scaled Jordan frames, and Schur-type majorization in hyperbolic systems

M. Seetharama Gowda , Juyoung Jeong , Sudheer Shukla This is my paper

Pith reviewed 2026-05-22 11:00 UTC · model grok-4.3

classification 🧮 math.OC

keywords hyperbolic polynomialsJordan frameshyperbolicity conesminimal polynomialsmajorizationEuclidean Jordan algebraseigenvalue mapsscaled Jordan frames

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The pith

For hyperbolic systems with a scaled Jordan frame, both the hyperbolic polynomial and its derivative are minimal polynomials generating their cones.

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

This paper proves that in a hyperbolic system of degree n at least 2 that has a scaled Jordan frame, the polynomial p and its first derivative p prime both qualify as minimal polynomials. A minimal polynomial generates the hyperbolicity cone of the system. The result extends an earlier theorem that required the cone to be generated by rank-one elements from the start. When the frame is normalized so each element has trace one and they sum to the direction vector e, the frame becomes orthonormal under the inner product coming from the eigenvalue map, the space includes a copy of real n-space as a Jordan algebra, and a majorization inequality of Schur type applies to associated doubly stochastic tuples.

Core claim

Corresponding to a hyperbolic system (V, p, e) with a scaled Jordan frame and degree n greater than or equal to 2, the polynomial p and the derivative polynomial p' are minimal polynomials that generate their respective hyperbolicity cones. This extends a result of Ito and Lourenço. For a Jordan frame where elements have trace one and sum to e, the frame is orthonormal with respect to the semi-inner product induced by the eigenvalue map λ, consists of exactly n elements, and V contains a copy of R^n as a Euclidean Jordan algebra. A Schur-type majorization result is also presented for a Jordan frame and an e-doubly stochastic n-tuple.

What carries the argument

The scaled Jordan frame, a finite collection of rank-one elements whose sum lies in the interior of the hyperbolicity cone.

If this is right

p generates the hyperbolicity cone for systems with scaled Jordan frames when n is at least 2
p prime generates the hyperbolicity cone of the derivative system under the same conditions
A Jordan frame is orthonormal relative to the semi-inner product induced by the eigenvalue map and has exactly n elements
V contains an embedded copy of R to the n as a Euclidean Jordan algebra
A Schur-type majorization inequality holds for Jordan frames paired with e-doubly stochastic n-tuples

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The minimality result may simplify the analysis of optimization problems defined over hyperbolicity cones
The link to Euclidean Jordan algebras suggests possible transfers of techniques between hyperbolic polynomial theory and semidefinite optimization
Existence of scaled Jordan frames might serve as a certificate for minimality in numerical examples of hyperbolic systems

Load-bearing premise

The hyperbolic system must admit at least one scaled Jordan frame, meaning a finite set of rank-one elements that sum to a point inside the hyperbolicity cone.

What would settle it

Construction of a hyperbolic polynomial of degree n greater than or equal to 2 together with a scaled Jordan frame for which p fails to generate the full hyperbolicity cone would disprove the main claim.

read the original abstract

Corresponding to a hyperbolic system $(V, p, e)$, where $V$ is a real finite-dimensional vector space and $p$ is a hyperbolic polynomial of degree $n$ in the direction $e$, we consider the eigenvalue map $\lambda: V \to R^n$ and the hyperbolicity cone $\Lambda_+$. In such a system, a scaled Jordan frame is defined as a finite set of rank-one elements whose sum lies in the interior of $\Lambda_+$. We show that when the system has a scaled Jordan frame and $n \geq 2$, $p$ and its derivative polynomial $p^\prime$ are minimal polynomials (generating their respective hyperbolicity cones), thereby extending a result of Ito and Louren{\c c}o proved in the setting of a rank-one generated (proper) hyperbolicity cone. When each element of a scaled Jordan frame has trace one and the total sum is $e$ (such a set is called a Jordan frame), we show that the frame is orthonormal relative to the semi-inner product induced by $\lambda$ with exactly $n$ elements, and $V$ contains a copy of $R^n$ (as a Euclidean Jordan algebra). We also present a Schur-type majorization result corresponding to a Jordan frame and an $e$-doubly stochastic $n$-tuple.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the Ito-Lourenço minimal polynomial result to scaled Jordan frames, with added orthonormality and Schur majorization when the frame sums to e.

read the letter

The punchline is that this work takes the Ito-Lourenço theorem on minimal polynomials for hyperbolicity cones and extends it from the rank-one generated proper cone case to the more general setting of systems that admit a scaled Jordan frame. It also establishes that the derivative polynomial is minimal for its cone, shows orthonormality for normalized frames, embeds a copy of Euclidean space as a Jordan algebra, and gives a Schur-type majorization result. What stands out as new is the move to scaled Jordan frames, which are finite sets of rank-one elements summing to an interior point. Under that assumption and for degree n at least 2, both p and p prime generate their cones as minimal polynomials. When the frame elements have trace one and sum to the direction e, the frame is orthonormal with respect to the inner product coming from the eigenvalue map, there are exactly n of them, and the space contains R to the n as a Euclidean Jordan algebra. The majorization then applies to e-doubly stochastic n-tuples. The paper handles this by working directly with the definitions of hyperbolic systems, the eigenvalue map, and the hyperbolicity cone. It avoids circularity by building on the existence of the frame rather than assuming extra structure. The citations to earlier work on hyperbolic polynomials and Jordan frames look appropriate and not over-relied upon in a self-referential way. One area that could use more attention is the prevalence of scaled Jordan frames in typical hyperbolic systems. The results are conditional on their existence, which may limit how often the conclusions apply in practice. Everything remains in finite dimensions over the reals, keeping the focus tight but also narrow. Since the full derivations are in the manuscript, a referee could check the steps for any subtle gaps in the extension. This paper targets researchers already working in hyperbolic programming, convex algebraic geometry, or optimization over symmetric cones. A reader who knows the Ito and Lourenço paper will see the precise differences and the added properties. It shows clear engagement with the literature and honest development of the ideas. I think it should go to peer review. The claims are specific enough that referees can evaluate them properly, and the extension adds something to the record even if the audience is specialized.

Referee Report

0 major / 0 minor

Summary. The manuscript studies hyperbolic systems (V, p, e) where p is a hyperbolic polynomial of degree n with direction e, together with the eigenvalue map λ and hyperbolicity cone Λ₊. It defines a scaled Jordan frame as a finite set of rank-one elements summing to an interior point of Λ₊. The central result asserts that, when such a frame exists and n ≥ 2, both p and its derivative p′ are minimal polynomials generating their respective cones; this extends the Ito-Lourenço theorem from the rank-one-generated case. For the special case of a Jordan frame (trace-one elements summing to e), the frame is shown to be orthonormal with respect to the λ-induced semi-inner product, V is shown to contain an embedded copy of ℝⁿ as a Euclidean Jordan algebra, and a Schur-type majorization inequality is derived for e-doubly stochastic n-tuples.

Significance. If the derivations are correct, the work meaningfully widens the setting in which minimality of hyperbolic polynomials can be asserted, replacing the restrictive rank-one-generated hypothesis with the existence of a scaled Jordan frame. The subsequent structural results on orthonormality, the Euclidean Jordan algebra embedding, and the majorization statement supply concrete algebraic tools that may prove useful in convex optimization over hyperbolic cones. The paper receives credit for stating the structural premise explicitly and for cleanly extending a known result without introducing internal circularity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. We are pleased that the extension of the Ito-Lourenço result via scaled Jordan frames, along with the orthonormality, Euclidean Jordan algebra embedding, and Schur-type majorization results, are viewed as meaningful contributions to the theory of hyperbolic systems.

Circularity Check

0 steps flagged

Derivation self-contained from structural assumption and external prior result

full rationale

The paper takes the existence of a scaled Jordan frame (finite set of rank-one elements summing to an interior point of the hyperbolicity cone) as an explicit hypothesis, together with n ≥ 2, and derives that both p and its derivative p' are minimal polynomials generating their respective cones. This extends the Ito-Lourenço theorem, which used the stronger assumption of a rank-one-generated proper cone. All subsequent claims (orthonormality of a trace-one Jordan frame, embedding of R^n as a Euclidean Jordan algebra, and Schur-type majorization) follow directly from the eigenvalue map λ and the given frame properties. No step reduces by construction to a fitted parameter, self-redefinition, or load-bearing self-citation chain; the argument is a standard implication from definitions and an external result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of a hyperbolic system (V, p, e) and the associated eigenvalue map and hyperbolicity cone; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond those already present in the cited literature on hyperbolic polynomials.

axioms (1)

domain assumption A hyperbolic polynomial p of degree n in direction e defines a hyperbolicity cone Λ+ and an eigenvalue map λ: V → R^n whose properties are taken from prior literature.
Invoked throughout the abstract as the ambient structure in which scaled Jordan frames are defined.

pith-pipeline@v0.9.0 · 5787 in / 1367 out tokens · 68454 ms · 2026-05-22T11:00:47.876350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that when the system has a scaled Jordan frame and n ≥ 2, p and its derivative polynomial p′ are minimal polynomials (generating their respective hyperbolicity cones), thereby extending a result of Ito and Lourenço.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

a scaled Jordan frame is defined as a finite set of rank-one elements whose sum lies in the interior of Λ+

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 2 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv 2004
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work page internal anchor Pith review Pith/arXiv arXiv 2005
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work page 2001

[2] [2]

Bhatia.Matrix Analysis

R. Bhatia.Matrix Analysis. Graduate Texts in Mathematics 169 (1997)

work page 1997

[3] [3]

Faraut and A

J. Faraut and A. Kor´ anyi.Analysis on Symmetric Cones. Oxford Uni- versity Press (1994)

work page 1994

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G˚ arding.An inequality for hyperbolic polynomials

L. G˚ arding.An inequality for hyperbolic polynomials. Journal of Math- ematics and Mechanics 8.6 (1959): 957-965

work page 1959

[5] [5]

M. S. Gowda.Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras. Linear Algebra and its Applications 528 (2017): 40-61

work page 2017

[6] [6]

Combinatorics hidden in hyperbolic polynomials and related topics

L. Gurvits.Combinatorics hidden in hyperbolic polynomials and related topics. arXiv preprint arXiv:math/0402088v1 (2004)

work page internal anchor Pith review Pith/arXiv arXiv 2004

[7] [7]

Combinatorial and algorithmic aspects of hyperbolic polynomials

L. Gurvits.Combinatorial and algorithmic aspects of hyperbolic polyno- mials. arXiv preprint arXiv:math/0404474v3 (2005)

work page internal anchor Pith review Pith/arXiv arXiv 2005

[8] [8]

J. W. Helton and V. Vinnikov.Linear matrix inequality representation of sets. Communications on Pure and Applied Mathematics 60.5 (2007): 654-674

work page 2007

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Ito and B

M. Ito and B. F. Louren¸ co.Automorphisms of rank-one generated hy- perbolicity cones and their derivative relaxations. SIAM Journal on Ap- plied Algebra and Geometry 7.1 (2023): 236-263

work page 2023

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A. W. Marshall, I. Olkin, and B. C. Arnold.Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics (2011)

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Nagano, B

T. Nagano, B. F. Louren¸ co, and A. Takeda.Projection onto hyperbol- icity cones and beyond: a dual Frank-Wolfe approach. arXiv preprint arXiv:2407.09213v3 (2025)

work page arXiv 2025

[12] [12]

Renegar.Hyperbolic programs, and their derivative relaxations

J. Renegar.Hyperbolic programs, and their derivative relaxations. Foun- dations of Computational Mathematics 6.1 (2006): 59-79

work page 2006

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Rudin.Functional Analysis

W. Rudin.Functional Analysis. McGraw-Hill, New York (1973). 30

work page 1973