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arxiv: 2606.23632 · v1 · pith:PAPBQFXBnew · submitted 2026-06-22 · 🧮 math.MG · cs.IT· math.IT

Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices

Tobias Boege , Ludovick Bouthat This is my paper

Pith reviewed 2026-06-26 05:37 UTC · model grok-4.3

classification 🧮 math.MG cs.ITmath.IT
keywords positive definite matricesprincipal minorsIngleton ratiosharp inequalitiesnonconvex optimizationsemialgebraic sets
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The pith

Closed-form solutions exist for a family of nonconvex optimization problems over the positive definite cone, yielding the exact infimum 16/27 for the Ingleton ratio in dimension 4.

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

The paper solves a family of nonconvex optimization problems that seek the greatest lower bounds on ratios of products of principal minors for real positive definite matrices. One direct consequence is an exact value of 16/27 for the infimum of the Ingleton ratio on 4 by 4 matrices, confirming an earlier conjecture. The same technique shows that the cone of all absolutely bounded such ratios fails to be polyhedral when the matrix size reaches 4 and is not semialgebraic over the rationals.

Core claim

Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over 4×4 positive definite matrices is 16/27, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for n≥4, and that it is not semialgebraic over Q.

What carries the argument

Closed-form solutions to nonconvex optimization problems over the positive definite cone, obtained by exploiting the algebraic structure of principal minors.

If this is right

  • The Ingleton ratio on 4 by 4 positive definite matrices is bounded below by exactly 16/27.
  • The cone of absolutely bounded ratios of principal-minor products is not polyhedral once the dimension is at least 4.
  • The same cone is not semialgebraic over the field of rational numbers.
  • Other members of the family of ratio-optimization problems over the positive definite cone likewise possess closed-form solutions.

Where Pith is reading between the lines

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

  • The algebraic approach may produce exact bounds for additional families of minor-product ratios in dimensions greater than 4.
  • The demonstrated failure of semialgebraicity over Q indicates that exact descriptions of the cone will generally require transcendental or non-rational coefficients.

Load-bearing premise

The family of nonconvex optimization problems over the positive definite cone admits analytically derivable closed-form solutions based on the algebraic structure of principal minors.

What would settle it

Exhibiting a single 4 by 4 positive definite matrix whose Ingleton ratio is strictly less than 16/27 would falsify the claimed infimum.

Figures

Figures reproduced from arXiv: 2606.23632 by Ludovick Bouthat, Tobias Boege.

Figure 1
Figure 1. Figure 1: The Wings of Ingleton — The gray convex body is the set of all positive definite 4 × 4 matrices of the form (16) in (t α, t, tβ )-space. The colored region consists of all points on which [φ] < 1. Points are colored on a linear gradient according to their value under [φ] from blue (close to 1) to red (close to 16/27). Lemma 4.6. For every Σ ∈ PD4, there exists Σ ′ ∈ PD4 of the form (17) Σ ′ =   1 a b c… view at source ↗
read the original abstract

We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript studies sharp inequalities for ratios of products of principal minors of real positive definite matrices. The main result provides closed-form solutions to a family of nonconvex optimization problems over the positive definite cone. As a special case, it proves that the infimum of the Ingleton ratio over 4×4 positive definite matrices is 16/27, confirming a conjecture of Hall and Johnson. It further shows that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for n≥4 and not semialgebraic over Q.

Significance. If the derivations hold, this work resolves an open conjecture with an explicit constant and provides structural results on the geometry of certain cones associated with principal minors. The closed-form solutions to the optimization problems represent a notable achievement in handling nonconvex problems over the PD cone. The demonstration that the cone is not polyhedral or semialgebraic adds to the understanding of its complexity. These results could impact areas like optimization, matrix analysis, and algebraic geometry.

minor comments (2)
  1. [Introduction] The definition of the Ingleton ratio could benefit from an explicit formula early in the paper to aid readers unfamiliar with the term.
  2. [§3] Some equations in the optimization section use notation that is introduced later; consider reordering for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external conjecture

full rationale

The paper's central claim is a closed-form solution to a family of optimization problems over the positive definite cone, with the Ingleton ratio infimum of 16/27 presented as confirmation of an external conjecture by Hall and Johnson. The abstract and available description frame the work as an independent algebraic result without reference to fitted parameters, self-definitional ratios, or load-bearing self-citations. No equations or derivation steps are supplied that reduce the claimed infimum or closed-form solutions to inputs by construction. Per the hard rules, absence of quotable reductions means the finding is no circularity (score 0).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definition and properties of the real positive definite cone and its principal minors; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The set of real positive definite matrices is an open convex cone in which all principal minors are positive.
    The optimization domain and the ratio functions are defined using this standard property from linear algebra.

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