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arxiv: 2604.27755 · v2 · pith:BMMOO7NNnew · submitted 2026-04-30 · 🧮 math.CO · math.AP· math.CA· math.OC· math.PR

G{aa}rding Polynomials

Hao Fang , Biao Ma This is my paper

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classification 🧮 math.CO math.APmath.CAmath.OCmath.PR
keywords Gårding polynomialsreal stable polynomialsRayleigh propertymatroidsnegative dependenceultra log-concavepolarizationaffine transformations
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The pith

Gårding polynomials extend real stable polynomials while preserving the Rayleigh property and ultra log-concavity for multi-affine nonnegative cases.

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

The paper defines Gårding polynomials by the geometric requirement that their positivity regions remain invariant when the input is translated by any positive vector and stay closed under strictly positive affine transformations. It proves this geometric condition is equivalent to reducing the polynomial to its multi-affine form via polarization and to a recursive test using partial derivatives. This construction properly contains the real stable polynomials yet keeps several of their useful features, including the Rayleigh property for multi-affine members with nonnegative coefficients and ultra log-concave sequences after positive univariate specialization. The property carries over to generating functions of many matroids under standard operations, which produces new negative dependence statements for matroid and graph classes that earlier real-stability techniques could not reach. The same framework supplies analogous conclusions for characteristic polynomials of selected matrix families.

Core claim

Gårding polynomials are defined by the requirement that their positivity regions are invariant under translation by positive vectors and closed under strictly positive affine transformations. The authors establish that this geometric formulation is equivalent both to polarization that reduces any such polynomial to the multi-affine case and to a recursion expressed through partial derivatives. The resulting class strictly contains the real stable polynomials while retaining key structural properties: multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations possess ultra log-concave coefficient sequences. The G

What carries the argument

the geometric invariance of positivity regions under positive-vector translations and strictly positive affine transformations, shown equivalent to polarization reduction and partial-derivative recursion

If this is right

  • Multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property.
  • Positive univariate specializations of these polynomials have ultra log-concave coefficient sequences.
  • The Gårding property is preserved for generating functions of several matroids under natural matroid operations.
  • New negative dependence results hold for generating functions associated with various matroid and graph classes previously outside real-stability methods.
  • Analogous negative dependence conclusions apply to characteristic polynomials arising from certain matrix classes.

Where Pith is reading between the lines

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

  • The geometric characterization may serve as a template for constructing larger polynomial classes in other areas of algebraic combinatorics where real stability is too restrictive.
  • One could check whether the Gårding property extends to generating functions of additional matroid families or to polynomials arising from hypergraphs.
  • Ultra log-concavity of the specializations may imply new concentration bounds for the probability measures supported on the associated combinatorial objects.

Load-bearing premise

The geometric definition of positivity-region invariance is equivalent to the polarization reduction to the multi-affine case and to the partial-derivative recursion.

What would settle it

An explicit polynomial whose positivity region satisfies the geometric invariance conditions but whose multi-affine form with nonnegative coefficients fails the Rayleigh property, or whose positive univariate specialization fails ultra log-concavity, would disprove the claimed preservation of those properties.

Figures

Figures reproduced from arXiv: 2604.27755 by Biao Ma, Hao Fang.

Figure 13.1
Figure 13.1. Figure 13.1: 3-connected rank 3 matroids with 6 elements 13.5. Conclusion of the proof. Proof of Theorem 12.3. The result follows by combining the closure properties from Proposi￾tions 13.6 and 13.11, Example 13.3, Theorem 13.4 and Theorem 13.12. □ Proof of Corollary 12.4. Let M be a matroid in one of the classes described in Theorem 12.3. We first consider the cospanning-set generating function. A subset X ⊆ E is c… view at source ↗
read the original abstract

We introduce G{\aa}rding polynomials, a class of real multivariate polynomials characterized by positivity regions that are invariant under translation by positive vectors and closed under strictly positive affine transformations. We prove that this geometric formulation is equivalent both to a reduction to the multi-affine setting via polarization and to a recursive criterion in terms of partial derivatives. The class of G{\aa}rding polynomials strictly extends that of real stable polynomials while preserving many of their structural properties. In particular, multi-affine G{\aa}rding polynomials with nonnegative coefficients satisfy the Rayleigh property, and their positive univariate specializations have ultra log-concave coefficient sequences. The G{\aa}rding property for several matroid generating functions is preserved under natural matroid operations. As applications, we derive new negative dependence results for generating functions associated with various classes of matroids and graphs, including examples previously beyond the scope of real stability and Lorentzian methods. We further obtain analogous results for characteristic polynomials arising from certain matrix classes.

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

1 major / 2 minor

Summary. The paper introduces Gårding polynomials as real multivariate polynomials whose positivity regions are invariant under translation by positive vectors and closed under strictly positive affine transformations. It proves equivalence of this geometric characterization both to a polarization reduction to the multi-affine setting and to a recursive criterion based on partial derivatives. The class strictly extends real stable polynomials while preserving structural properties: multi-affine Gårding polynomials with nonnegative coefficients satisfy the Rayleigh property, and positive univariate specializations have ultra log-concave coefficient sequences. The Gårding property is shown to be preserved under natural matroid operations, yielding new negative dependence results for generating functions of various matroid and graph classes (including some previously outside real stability and Lorentzian methods) as well as for characteristic polynomials of certain matrix classes.

Significance. If the equivalences hold, the work meaningfully enlarges the scope of stability-type techniques in combinatorial algebra and matroid theory. It supplies a new framework for establishing negative dependence and log-concavity results for matroid classes that real stability and Lorentzian polynomials do not reach, with concrete applications to graphic matroids, transversal matroids, and matrix-based constructions. The explicit preservation statements under deletion, contraction, and other operations add immediate combinatorial utility.

major comments (1)
  1. [§3, Theorem 3.4] §3, Theorem 3.4 (equivalence of geometric invariance to polarization and partial-derivative recursion): the argument that the affine-closure condition implies the recursive criterion without presupposing real stability or additional sign hypotheses on the coefficients is load-bearing for the strict-extension claim; the current write-up does not explicitly isolate the step that avoids circularity with real-stability assumptions, and a counter-example check or additional lemma for non-stable cases would strengthen the foundation for the matroid applications in §5.
minor comments (2)
  1. [§2] Notation for the positivity region (e.g., the set denoted P_f) is introduced in §2 but used without re-statement in the statements of Theorems 4.2 and 5.1; a brief reminder or cross-reference would improve readability.
  2. [Corollary 4.7] The ultra-log-concavity claim for univariate specializations (Corollary 4.7) would benefit from an explicit statement of the degree and positivity hypotheses under which the coefficient sequence is considered.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. The major comment on Theorem 3.4 raises a valid point about clarity in the proof structure. We address it point by point below and have revised the manuscript to strengthen the exposition.

read point-by-point responses

  1. Referee: [§3, Theorem 3.4] §3, Theorem 3.4 (equivalence of geometric invariance to polarization and partial-derivative recursion): the argument that the affine-closure condition implies the recursive criterion without presupposing real stability or additional sign hypotheses on the coefficients is load-bearing for the strict-extension claim; the current write-up does not explicitly isolate the step that avoids circularity with real-stability assumptions, and a counter-example check or additional lemma for non-stable cases would strengthen the foundation for the matroid applications in §5.

    Authors: We agree that isolating this implication is essential for the strict-extension claim. In the proof of Theorem 3.4, the direction from geometric invariance (positivity region invariant under positive translations and closed under strictly positive affine maps) to the partial-derivative recursion is established directly from the definitions, without any appeal to real stability or coefficient sign conditions. The argument proceeds by applying a suitable positive affine transformation to reduce to the multi-affine case via polarization, then verifying the recursive criterion on the resulting polynomials using only the invariance properties. To address the referee's concern explicitly, we have added a new remark immediately after the proof that isolates this step, notes its independence from stability assumptions, and includes a brief verification on a concrete non-stable Gårding polynomial (a bivariate example outside the real stability cone but satisfying the geometric conditions). We have also incorporated the suggested counter-example check. These changes clarify the foundation without altering the theorem statement and directly support the matroid applications in §5. revision: yes

Circularity Check

0 steps flagged

No circularity: geometric definition proved equivalent to polarization and recursion without reduction to inputs

full rationale

The paper defines Gårding polynomials geometrically via invariance of positivity regions under positive translations and strictly positive affine transformations. It then proves this is equivalent to polarization reduction to multi-affine case and to a partial-derivative recursion. This is presented as a theorem establishing equivalence rather than a definitional loop or fitted input renamed as prediction. No self-citations are invoked as load-bearing for the core equivalence, and the extension to matroid applications follows from the proved properties (Rayleigh, ultra-log-concavity) without reducing derived quantities back to the original fit or definition by construction. The derivation chain remains self-contained against external notions of stability.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the equivalence between the geometric definition and the two algebraic characterizations (polarization and partial derivatives). No free parameters are introduced. The work assumes standard facts from real algebraic geometry and matroid theory but does not postulate new entities with independent evidence.

axioms (1)
  • domain assumption Equivalence of the geometric positivity-region definition to the polarization reduction and the partial-derivative recursion holds for real multivariate polynomials.
    This equivalence is invoked to transfer properties from the multi-affine case to the general case and is the step that justifies calling the class an extension of real stable polynomials.

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