Lorentzian polynomials

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#Lorentzian polynomials

P. Brändén and J. Huh introduced the notion of Lorentzian polynomials [BH20]. All stable polynomials are Lorentzian, but the converse is not true for polynomials of degree three or more. For an introduction to the topic, see these lectures by Brändén.

Definition (Lorentzian polynomial).

A homogeneous polynomial $h(x_1,\dotsc,x_n)$ of degree $d$ is strictly Lorentzian if

All coefficients of $h$ are positive, and
The signature of (the degree $2$ polynomial) \[\frac{\partial}{\partial x_{i_1}}\frac{\partial}{\partial x_{i_2}} \dotsm \frac{\partial}{\partial x_{i_e}} h(x_1,\dotsc,x_n)\] is $(+,-,-,\dotsc,-)$ for any $i_1,\dotsc,i_e \in [n]$ where $e = d-2.$

To compute the signature of a homogeneous polynomial $P(x_1,\dotsc,x_n)$ of degree 2, consider the symmetric matrix \[\left( \frac{\partial}{\partial x_{i}}\frac{\partial}{\partial x_{j}} P \right)_{ij} \in \setR^{n \times n}.\] If it has only one positive eigenvalue, while the remaining are negative, then it has signature $(+,-,-,\dotsc,-).$

A polynomial is called Lorentzian if it is the limit of strictly Lorentzian polynomials. There are two other equivalent conditions listed in [HMMD22].

An equivalent definition for Lorentzian is:

All coefficients are non-negative.
The support of the polynomial forms an M-convex set.
All quadratic forms (as above) has at most one positive eigenvalue.

The basis-generating polynomial of a matroid is Lorentzian. More generally, integral polymatroids give Lorentzian normalized support polynomials. For the support condition, saturated Newton polytopes, and discrete convexity background, see Newton polytopes and $M$-convexity.

Determining if polynomials are Lorentzian can be done efficiently, see [Chi24]. This is in contrast with checking for real stability which is coNP-complete, see [Chi24].

The following operations preserve the Lorentzian property:

Taking the multi-affine part of a Lorentzian polynomial.
Products of Lorentzian polynomials are Lorentzian.
Specializing variables in a multiaffine Lorentzian polynomial.

In [HMMD22], several generalizations of Schur polynomials are either proved or conjectured to be Lorentzian.

Theorem (See [HMMD22]).

For any integer partition $\lambda,$ the normalized Schur polynomial \[\sum_{ \mu } K_{\lambda \mu} \frac{\monomial_\mu}{\mu_1! \mu_2 ! \dotsm \mu_\ell !}\] is Lorentzian. Here, $K_{\lambda \mu}$ are the Kostka coefficients.

Another proof of this is given in [RT23].

A. Khare, J. P. Matherne, and Avery St. Dizier extend this representation-theoretic picture to shifted characters of parabolic Verma modules over $\mathfrak{sl}_{n+1}(\setC)$ [KMD25]. More generally, they show that shifted and normalized characters of the Kostant partition function of any loopless multigraph on $[n+1]$ are Lorentzian, and they give limits on how far such log-concavity phenomena can be extended.

The skew Schur case of this conjecture is now a theorem. If $f=\sum_\alpha c_\alpha\xvec^\alpha,$ write \[\mathcal{N}(f) \coloneqq \sum_\alpha\frac{c_\alpha}{\alpha!}\xvec^\alpha.\]

Theorem (Zhang, [Zha26]).

For partitions $\mu\subseteq\lambda$ and $n\geq1,$ the normalized skew Schur polynomial \[\mathcal{N}\bigl( \schurS_{\lambda/\mu}(x_1,\dotsc,x_n) \bigr)\] is Lorentzian.

The central step comes from Schubert polynomials. Every finite-variable skew Schur polynomial is a zero-specialization of a Schubert polynomial indexed by a stabilized $321$-avoiding permutation. Schubert polynomials are dually Lorentzian, and that property survives the specialization. A rectangular-complement identity then turns dual Lorentzianity into Lorentzianity of the normalization.

K.-H. Nguyen-Dang independently proves the stronger geometric statement that normalized skew Schur polynomials are realizable volume polynomials over $\setC$ [Ngu26]. The same result holds for skew Schur $P$- and $Q$-polynomials, and nonzero instances admit smooth irreducible projective realizations with semiample divisors.

Here, a homogeneous polynomial of degree $d$ in $x_1,\dotsc,x_n$ is a realizable volume polynomial over a field if it is a nonnegative rational multiple of the volume polynomial \[(x_1,\dotsc,x_n) \mapsto \frac{1}{d!}\int_X (x_1 D_1 + \dotsb + x_n D_n)^d\] of semiample Cartier divisor classes $D_1,\dotsc,D_n$ on a $d$-dimensional integral projective variety $X$ over that field. Over $\setC,$ semiample classes are nef, so every realizable volume polynomial is Lorentzian [Thm. 4.6, BH20].

Problem

Find a direct combinatorial proof that normalized skew Schur polynomials are Lorentzian. The known proofs use Schubert geometry or volume realizations rather than a tableau-level Lorentzian argument.

The corresponding questions for key polynomials and ordinary Schubert polynomials were settled in a stronger geometric form by K.-H. Nguyen-Dang and Z. Wang.

Theorem (Nguyen-Dang–Wang, [Thm. 1.1, NW26]).

Let $\alpha$ be a weak composition and let $w\in\symS_n.$ Then $\mathcal{N}(\key_\alpha),$ $\mathcal{N}(\atom_\alpha),$ and $\mathcal{N}(\schubert_w)$ are realizable volume polynomials over every field. The same conclusion holds for the factorial normalization of every sign-corrected homogeneous component of $\grothendieck_w$ and for every homogeneous Lascoux and Lascoux-atom layer.

Over $\setC,$ every nonzero polynomial in this list is Lorentzian.

The construction first realizes the corresponding homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets; the polynomials in the theorem occur as volume minors of these packets. This proves Conjectures 15, 21, 22, and 23 of [HMMD22]. It also proves the relevant saturated-Newton-polytope conjectures for Demazure atoms and for ordinary Grothendieck, Lascoux, and Lascoux-atom polynomials [MTY19]. The result concerns ordinary single-alphabet type $A$ polynomials, not arbitrary double Schubert or double Grothendieck polynomials.

Earlier, 0-1-Grothendieck polynomials were shown to be Lorentzian in [CFY24].

In [CQ25], the authors study the set of symmetric polynomials which are also Lorentzian. This set is homeomorphic to a closed Euclidean ball. They also give a reduction scheme for testing Lorentzianity of symmetric polynomials, semialgebraic descriptions in small degrees, and examples showing that some natural symmetric operators do not preserve the Lorentzian property.

Example (Chromatic symmetric polynomials).

For abelian Dyck paths, the chromatic symmetric polynomial (in any number of variables) is Lorentzian, see [MMS22].

In contrast, there is a (non-abelian) unit interval graph for which the chromatic symmetric polynomial is not Lorentzian, or normalized Lorentzian. The area sequence for when this fails is $01121121$ — this example was found by R.I. Liu and C. Vinzant.

In [RSW23], the authors describe a way to construct operators preserving the Lorentzian property.

See [BL23, Ros23] for Lorentzian polynomials on fans. See [HX23] for Lorentzian polynomials and intersection theory.

For Lorentzian polynomials from independence sets in graphs, see [BH25]. J. Lee, J. Oh, and J. Seo use Lorentzian polynomial methods to prove homomorphism inequalities for antiferromagnetic graphs [LOS25]. This includes progress on coloring and independent-set extremal problems for regular graphs.

Bibliography

[BH25]Amire Bendjeddou and Leonard Hardiman. Lorentzian polynomials and the independence sequences of graphs. Bulletin of the London Mathematical Society, 57(4):1305–1323, February 2025.

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@article{BendjeddouHardiman2025,
  title = {Lorentzian polynomials and the independence sequences of graphs},
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  journal = {Bulletin of the London Mathematical Society},
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  pages = {1305--1323}
}

[BH20]Petter Brändén and June Huh. Lorentzian polynomials. Annals of Mathematics, 192(3), November 2020.

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  publisher = {Annals of Mathematics},
  author = {Br{\"{a}}nd{\'{e}}n,  Petter and Huh,  June},
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}

[BL23]Petter Brändén and Jonathan Leake. Lorentzian polynomials on cones. arXiv:2304.13203, 2023.

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@article{BrandenLeake2023x,
Author = {Petter Br{\"{a}}nd{\'{e}}n and Jonathan Leake},
Title = {Lorentzian polynomials on cones},
Year = {2023},
Eprint = {2304.13203},
  url = {https://arxiv.org/abs/2304.13203},
journal = {arXiv e-prints}
}

[CFY24]Yiming Chen, Neil J. Y. Fan and Zelin Ye. Zero‐one Grothendieck polynomials. arXiv:2405.05483, 2024.

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@article{ChenFanYe2024x,
  author    = {Yiming Chen and Neil J.~Y. Fan and Zelin Ye},
  title     = {Zero‐one {G}rothendieck Polynomials},
  journal   = {arXiv preprint},
  eprint    = {2405.05483},
  year      = {2024},
  archivedAt = {arXiv},
  url       = {https://arxiv.org/abs/2405.05483}
}

[Chi24]Tracy Chin. Real stability and log concavity are coNP-complete. arXiv:2405.00162, 2024.

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@article{Chin2024x,
Author = {Tracy Chin},
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Year = {2024},
Eprint = {2405.00162},
  url = {https://arxiv.org/abs/2405.00162},
journal = {arXiv e-prints}
}

[CQ25]Tracy Chin and Daniel Qin. Symmetric Lorentzian polynomials. arXiv:2510.07819v1, 2025.

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@article{ChinQin2025x,
  Author        = {Tracy Chin and Daniel Qin},
  Title         = {Symmetric {L}orentzian Polynomials},
  Year          = {2025},
  Eprint        = {2510.07819v1},
  url = {https://arxiv.org/abs/2510.07819v1},
  journal       = {arXiv e-prints}
}

[HX23]Jiajun Hu and Jian Xiao. Intersection theoretic inequalities via Lorentzian polynomials. arXiv:2304.04191, 2023.

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@article{HuXiao2023x,
Author = {Jiajun Hu and Jian Xiao},
Title = {Intersection theoretic inequalities via {L}orentzian polynomials},
Year = {2023},
Eprint = {2304.04191},
  url = {https://arxiv.org/abs/2304.04191},
journal = {arXiv e-prints}
}

[HMMD22]June Huh, Jacob Matherne, Karola Mészáros and Avery St. Dizier. Logarithmic concavity of Schur and related polynomials. Transactions of the American Mathematical Society, 375(6):4411–4427, March 2022.

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  url2 = {https://doi.org/10.1090/tran/8606},
  year = {2022},
  month = mar,
  publisher = {American Mathematical Society ({AMS})},
  volume = {375},
  number = {6},
  pages = {4411--4427},
  author = {June Huh and Jacob Matherne and Karola M{\'{e}}sz{\'{a}}ros and Avery St. Dizier},
  title = {Logarithmic concavity of {S}chur and related polynomials},
  journal = {Transactions of the American Mathematical Society}
}

[KMD25]Apoorva Khare, Jacob P. Matherne and Avery St. Dizier. Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions. arXiv:2504.01623, 2025.

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@article{KhareMatherneDizier2025x,
  author = {Apoorva Khare and Jacob P. Matherne and Avery St. Dizier},
  title = {Log-concavity of characters of parabolic {V}erma modules, and of restricted {K}ostant partition functions},
  year = {2025},
  eprint = {2504.01623},
  url = {https://arxiv.org/abs/2504.01623},
  journal = {arXiv e-prints}
}

[LOS25]Joonkyung Lee, Jaeseong Oh and Jaehyeon Seo. Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials. arXiv:2506.13659, 2025.

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@article{LeeOhSeo2025x,
  author = {Joonkyung Lee and Jaeseong Oh and Jaehyeon Seo},
  title = {Counting homomorphisms in antiferromagnetic graphs via {L}orentzian polynomials},
  year = {2025},
  eprint = {2506.13659},
  url = {https://arxiv.org/abs/2506.13659},
  journal = {arXiv e-prints}
}

[MMS22]Jacob P. Matherne, Alejandro H. Morales and Jesse Selover. The Newton polytope and Lorentzian property of chromatic symmetric functions. arXiv:2201.07333, 2022.

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@article{MatherneMoralesSelover2022x,
Author = {Jacob P. Matherne and Alejandro H. Morales and Jesse Selover},
Title = {The {N}ewton polytope and {L}orentzian property of chromatic symmetric functions},
Year = {2022},
Eprint = {2201.07333},
  url = {https://arxiv.org/abs/2201.07333},
journal = {arXiv e-prints}
}

[MTY19]Cara Monical, Neriman Tokcan and Alexander Yong. Newton polytopes in algebraic combinatorics. Selecta Mathematica, 25(5), October 2019.

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  number = {5},
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  publisher = {Springer Science and Business Media LLC},
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[Ngu26]Khai-Hoan Nguyen-Dang. Richardson volume models for skew Schur and skew Schur ${P}/{Q}$-functions. arXiv:2608.10516v1, 2026.

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@article{NguyenDang2026x,
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    ${P}/{Q}$-functions},
  year = {2026},
  eprint = {2608.10516v1},
  url = {https://arxiv.org/abs/2608.10516v1},
  journal = {arXiv e-prints}
}

[NW26]Khai-Hoan Nguyen-Dang and Zhenpeng Wang. Canonical-row Chern flow on Bott–Samelson towers: Realizable-volume models for Schubert, Grothendieck, and Lascoux polynomials. arXiv:2609.02850v1, 2026.

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}

[Ros23]Dustin Ross. Lorentzian fans. arXiv:2304.13176, 2023.

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@article{Ross2023x,
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Year = {2023},
Eprint = {2304.13176},
  url = {https://arxiv.org/abs/2304.13176},
journal = {arXiv e-prints}
}

[RSW23]Julius Ross, Hendrik Süß and Thomas Wannerer. Dually Lorentzian polynomials. arXiv:2304.08399, 2023.

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@article{RossSussWannerer2023x,
Author = {Julius Ross and Hendrik Süß and Thomas Wannerer},
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  url = {https://arxiv.org/abs/2304.08399},
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}

[RT23]Julius Ross and Matei Toma. On Hodge-Riemann cohomology classes. Birational geometry, Kähler–Einstein metrics and degenerations:763–793, May 2023.

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	journal = {SpringerLink},
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	year = {2023},
	month = may,
	issn = {2194-1017},
	isbn = {978-3-031-17859-7},
	publisher = {Springer},
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}

[Zha26]Philip B. Zhang. Normalized skew Schur polynomials are Lorentzian. arXiv:2608.12266v1, 2026.

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@article{Zhang2026x,
  author = {Philip B. Zhang},
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  url = {https://arxiv.org/abs/2608.12266v1},
  journal = {arXiv e-prints}
}