Lorentzian Polynomials

• Lorentzian polynomials are homogeneous polynomials with nonnegative coefficients that satisfy a Hessian signature condition and M-convex support, ensuring strong log-concavity.
• They connect algebraic geometry, matroid theory, and optimization by generalizing convexity concepts and encoding combinatorial inequalities and negative dependence.
• Applications include capacity bounds, stability analysis in dynamical systems, and combinatorial optimization, underpinned by rigorous Hessian and derivative conditions.

A Lorentzian polynomial is a homogeneous polynomial in several variables, typically with nonnegative coefficients, whose analytic and combinatorial properties encode strong forms of log-concavity and negative dependence. The Lorentzian condition captures a Hessian signature constraint, generalizes notions from convex and tropical geometry, and naturally interacts with matroid theory, optimization, algebraic geometry, and stability theory. The theory is deeply connected with structures such as M-convex sets, Chow rings, Hodge–Riemann relations, and tropical hyperfields, and supports applications including combinatorial inequalities, capacity bounds, and dynamical stability in convex cones.

1. Definition and Algebraic Characterization

A homogeneous polynomial $f(w_1, ..., w_n)$ of degree $d$ with nonnegative coefficients is Lorentzian if two main properties hold:

• Support and M-convexity: The exponent vectors $\alpha$ of monomials with nonzero coefficient form an M-convex set (discrete convexity; in degree $d$, the set lies in the discrete simplex $\Delta^d_n$).
• Hessian signature: For every partial derivative of $f$ of order $d-2$, the associated quadratic form (its Hessian) at any point of the positive orthant has exactly one positive eigenvalue ("Lorentzian signature") (Brändén et al., 2019). For general degree $d \geq 3$, the condition is recursive: directional derivatives of any order must yield a quadratic polynomial whose Hessian preserves this signature.

This is equivalent to strong log-concavity or complete log-concavity (all directional derivatives remain log-concave), and is closely tied to Hodge–Riemann relations: the "one positive eigenvalue" criterion reflects Lefschetz-type phenomena in algebraic geometry.

For Lorentzian polynomials on proper convex cones $\mathcal{K}$ (or open convex cones $\omega$), definitions generalize: all $d$-fold directional derivatives in the interior of $\mathcal{K}$ must be positive and all Hessians must have Lorentzian signature (Brändén et al., 2021, Brändén et al., 2023, Blekherman et al., 21 May 2024, Dey, 4 Jan 2025). In the context of self-dual cones, the associated quadratic form must be $\mathcal{K}$-positive, tightly connecting to generalized Perron–Frobenius theory.

2. Connections to Matroids and M-Convexity

• The support of any Lorentzian polynomial is an M-convex subset. In the multi-affine case, this is equivalent to the set of bases of a matroid.
• For matroids, the generating function (sum over bases) is Lorentzian if and only if the base set is M-convex (Brändén et al., 2019, Baker et al., 4 Aug 2025).
• Representations over triangular hyperfields $\mathbb{T}_q$ (tropical and degenerate limits) yield a geometric realization of the orbit spaces of Lorentzian polynomials (Baker et al., 4 Aug 2025). The projectivized space $\mathbb{P}L_J$ of Lorentzian polynomials with support $J$ aligns with thin Schubert cells in the tropical setting.
• Negative dependence phenomena, ultra log-concavity (ULC), and Rayleigh properties are encoded via the Lorentzian structure, providing routes to Mason’s ultra log-concavity conjecture and negative correlation results.

3. Geometric and Topological Structure

• Spaces of Lorentzian polynomials (modulo rescaling) are topological manifolds with boundary, of dimension equal to the Tutte rank of the support $J$ (Baker et al., 4 Aug 2025). In favorable cases ("rigid" matroids), their Hausdorff compactification is homeomorphic to a closed Euclidean ball.
• Compactifications correspond to moduli spaces such as the Chow quotient of the Grassmannian and, in dimension two, to the moduli of stable rational curves $\overline{M}_{0,n}$ (Baker et al., 4 Aug 2025). The Dressian, or tropical Grassmannian, manifests as part of the boundary of these spaces.
• There are explicit homeomorphisms between the space of Lorentzian polynomials and moduli of matroid representations over (tropical) hyperfields for various matroids (e.g., uniform, Betsy Ross, elliptic).
• In certain cases, the closure of $\mathbb{P}L_J$ is not a ball (e.g., the elliptic matroid $T_{11}$ or for stable polynomials in $B_{11}$), with computed Euler characteristics confirming non-trivial topology.

4. Lorentzian Polynomials on Cones and Generalizations

• Lorentzian polynomials can be defined over arbitrary proper convex cones $\mathcal{K}$ as $\mathcal{K}$-Lorentzian polynomials (Blekherman et al., 21 May 2024, Dey, 4 Jan 2025). The central analytic requirement is that all appropriate quadratic forms derived via directional derivatives have exactly one positive eigenvalue and induce a strong positivity condition over $\mathcal{K}$.
• There is an equivalence between the class of $\mathcal{K}$-Lorentzian polynomials and $\mathcal{K}$-completely log-concave (CLC) polynomials (Blekherman et al., 21 May 2024, Dey, 4 Jan 2025). Complete log-concavity translates into the Lorentzian Hessian signature condition at all relevant points.
• As the cone $\mathcal{K}$ varies, especially in settings like the positive semidefinite cone $S_+^n$, algorithmic complexity arises in recognizing Lorentzian polynomials, with NP-hardness in the quartic case (Blekherman et al., 21 May 2024).

5. Capacity, Inequalities, and Combinatorial Applications

• Capacity inequalities for Lorentzian polynomials generalize classical bounds, such as Newton’s inequalities, to the multivariate setting (Schweitzer, 2020). For a polynomial $P$ and a multi-index $\alpha$, $\text{Cap}_\alpha(P)$ expresses a measure of extremal coefficient growth, and rigorous probabilistic methods connect these bounds to binomial distributions.
• Nikolskii-type and Markov-type inequalities are sharp for Lorentz polynomials, controlling $L^p$ norms and derivative maxima in terms of polynomial norms, with constants improved over Erdős-type bounds for polynomials with zero constraints (Erdelyi, 2014).
• In graph theory, transformed independence polynomials become Lorentzian after suitable homogenization, which yields ultra log-concavity and implies unimodality for independence sequences of large classes of graphs (e.g., those produced by the $R_{W_4}$ operator), making progress on longstanding combinatorial conjectures (Bendjeddou et al., 1 May 2024).

6. Lorentzian Signature, Hyperbolicity, and Algebraic Implications

• The Lorentzian signature condition of the Hessian (exactly one positive eigenvalue) connects this theory to that of hyperbolic polynomials, conic stability, and generalized Hodge–Riemann relations. Hyperbolic programming exploits these curvature properties for efficient computation (such as approximating the permanent of matrices) (Dey, 2022).
• Dually Lorentzian polynomials induce operators that preserve the Lorentzian property, enabling generalized Alexandrov–Fenchel inequalities for volumes, mixed discriminants, and convex body invariants (Ross et al., 2023).
• Lorentzian polynomials derive from or induce hereditary Lorentzian polynomials on cones, with combinatorial connectivity conditions (via simplicial complexes) encoding the algebraic/differential structure directly (Brändén et al., 2023).

7. Interactions with Algebraic Geometry and Projective Varieties

• Volume polynomials arising from convex bodies or as degree polynomials of projective varieties (using nef divisors) are Lorentzian (Brändén et al., 2019, An et al., 3 Dec 2024). This includes the degree polynomials for Richardson varieties and dual Schubert polynomials, establishing log-concavity, M-convexity, and combinatorial saturation for their supports.
• Chow rings and intersecting properties of simplicial fans can be characterized in terms of hereditary Lorentzian polynomials, with the Hodge–Riemann relations of degree zero and one captured by the Lorentzian signature (Brändén et al., 2023, Brändén et al., 2021).
• Postnikov–Stanley polynomials are Lorentzian, and their support is a generalized permutahedron (M-convex), resolving previous conjectures (An et al., 3 Dec 2024).

8. Stability Theory and Dynamical Applications

• $\mathcal{K}$-Lorentzian polynomials (or $\mathcal{K}$-CLC forms) characterize Hurwitz stability, particularly for degrees $d \leq 4$ and under explicit coefficient bounds for higher degrees (Dey, 4 Jan 2025). This has direct implications for stability analysis in dynamical systems, including evolution variational inequalities (EVIs) where trajectories are constrained to convex sets.
• The concept of an associated "cone" $\mathcal{K}(f,v)$ for a polynomial $f$ and direction $v$ provides a proper domain for log-concavity and stability characterization, extending classical hyperbolicity cones.

9. Summary Table: Core Definitions and Properties

Concept	Analytic/Algebraic Property	Combinatorial/Geometric Property
Lorentzian Poly.	Hessian: $\leq$ 1 positive eigenvalue	Support is M-convex
Volume Poly.	Satisfies Alexandrov–Fenchel	Subset of Lorentzian poly.
$\mathcal{K}$-Lorentzian	Signature, positivity over cone $\mathcal{K}$	Generalizes nonneg. orthant case
Dually Lorentzian	Operators preserve Lorentzian class	Generalized AF inequalities
Hereditary Lorentzian	Connectivity of simplicial complex + Hessian	For Chow rings/simplicial fans

10. Open Questions and Future Directions

• Topological classification of Lorentzian polynomial spaces for various matroids and their compactifications, including cases with non-ball-like closures (Baker et al., 4 Aug 2025).
• Full extension and characterization of hereditary Lorentzian polynomials to higher codimension and general variables (Marques et al., 2022).
• Algorithmic recognition and efficient computation (e.g., of coefficients, capacity, permanents) in broader cones and for higher-order derivatives.
• Extension of stability theory to more general dynamical systems using $\mathcal{K}$-Lorentzian analysis.

Lorentzian polynomials thus unify and extend foundational results in discrete convexity, algebraic geometry, optimization, and combinatorics by encoding log-concavity, negative dependence, and deep topological and geometric structure across these domains.