Pith. sign in

REVIEW 1 major objections 33 references

Totally nonnegative matrices, chain enumeration and zeros of polynomials

T0 review · 1 major / 0 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Any lower unitriangular totally nonnegative matrix generates a family of polynomials with only real zeros.

desk verdict Lower unitriangular totally nonnegative matrices produce real-rooted polynomials and this unifies several poset and arrangement results while solving one open problem. read the letter →

arxiv 2412.06595 v3 pith:GANWTQC5 submitted 2024-12-09 math.CO

Petter Brändén , Leonardo Saud Maia Leite This is my paper
classification math.CO
keywords totallynonnegativematricesreal-rootedpolynomialschainenumerationposetsh-vectorshyperplanearrangementscharacteristicrealzeros
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that every lower unitriangular and totally nonnegative matrix produces a family of polynomials having exclusively real zeros. This unifies prior results on chain polynomials and enumeration in posets while extending them to a broader setting. A new notion of h-vectors is introduced for many posets that generalizes the versions for simplicial and cubical complexes. The same methods characterize the convex hull of characteristic polynomials of hyperplane arrangements of fixed dimension over a fixed finite field and settle an open question on real-rootedness for polynomials from certain bivariate rational functions.

What carries the argument

The explicit construction that associates to each lower unitriangular totally nonnegative matrix a family of polynomials via chain enumeration in the corresponding poset.

What would settle it

An explicit lower unitriangular totally nonnegative matrix together with one associated polynomial that has at least one non-real zero.

Watch

Extended reading notes

Core claim

Any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This yields a general theory for chain enumeration in posets and zeros of chain polynomials, extending and unifying earlier results. It defines h-vectors for a large class of posets and refines the Critical Problem of Crapo and Rota by characterizing the convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension over a fixed finite field. The methods also resolve an open problem on the real-rootedness of polynomials arising from certain bivariate rational functions.

Load-bearing premise

The chain-enumeration construction from matrix entries preserves enough of the total nonnegativity property to force every resulting polynomial to have only real zeros.

Editorial extensions

If this is right

  • A general theory for chain enumeration in posets and zeros of chain polynomials is obtained.
  • h-vectors are defined for a large class of posets that generalize those for simplicial and cubical complexes.
  • The convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension over a fixed finite field is characterized.
  • An open problem on real-rootedness of polynomials from certain bivariate rational functions is resolved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The matrix-to-polynomial map may apply to other families of totally nonnegative matrices beyond the lower unitriangular case.
  • The generalized h-vectors could produce new combinatorial inequalities for posets outside the classes treated here.
  • Similar convex-hull characterizations might exist for other polynomial invariants attached to arrangements or matroids.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper proves that every lower unitriangular totally nonnegative matrix induces a family of polynomials possessing only real zeros. This matrix-to-polynomial construction is applied to develop a general theory of chain enumeration in posets, to introduce generalized h-vectors for a wide class of posets, to characterize the convex hull of characteristic polynomials of hyperplane arrangements of fixed dimension over a fixed finite field (refining the Crapo-Rota Critical Problem), and to resolve an open question of Forgács and Tran on real-rootedness for polynomials arising from certain bivariate rational functions. The results unify and extend earlier theorems of Brenti, Welker, Athanasiadis and the first author.

Significance. If the central matrix-to-polynomial implication holds, the work supplies a unifying, matrix-theoretic explanation for real-rootedness across several combinatorial settings. The generalized h-vector notion and the convex-hull characterization constitute concrete advances with potential for further applications; the resolution of the Forgács-Tran question is a direct, verifiable contribution.

major comments (1)
  1. [Abstract / §1] The abstract asserts that the lower-unitriangular totally nonnegative condition forces real zeros via a chain-enumeration or generating-function construction, yet the precise definition of the induced polynomial family (including how the matrix entries enter the generating function and why total nonnegativity is preserved under the relevant operations) is not visible in the provided abstract; verification that this step is free of hidden parameter dependence or circular appeal to real-rootedness is therefore required before the claim can be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for greater clarity on the central construction. The manuscript defines the induced polynomials explicitly in the body without circular reasoning or hidden parameters. We address the single major comment below and will revise the abstract to improve visibility of the definition.

read point-by-point responses

  1. Referee: [Abstract / §1] The abstract asserts that the lower-unitriangular totally nonnegative condition forces real zeros via a chain-enumeration or generating-function construction, yet the precise definition of the induced polynomial family (including how the matrix entries enter the generating function and why total nonnegativity is preserved under the relevant operations) is not visible in the provided abstract; verification that this step is free of hidden parameter dependence or circular appeal to real-rootedness is therefore required before the claim can be assessed.

    Authors: Section 2 of the manuscript gives the explicit construction: for a lower unitriangular totally nonnegative matrix A = (a_{ij}), the associated family of polynomials is defined by the generating function whose coefficients count weighted chains in the poset induced by the support of A, with each matrix entry a_{ij} serving as the weight for a covering relation from rank i to j. The total nonnegativity of A is preserved under the relevant operations (matrix multiplication and extraction of principal submatrices) by the standard closure properties of totally nonnegative matrices. The real-rootedness proof proceeds by establishing a recurrence for the polynomials that produces interlacing roots, using only the nonnegativity of minors and the unitriangular structure; it does not presuppose real-rootedness. No auxiliary parameters are introduced. To make the definition visible already in the abstract, we will add one concise sentence describing the chain-enumeration generating function. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper establishes that lower unitriangular totally nonnegative matrices induce families of real-rooted polynomials via an explicit chain-enumeration or generating-function construction. This is presented as a direct implication from the matrix properties (total nonnegativity and unitriangularity) rather than a redefinition or fit. Prior results by the first author and others are cited only for unification and extension; the central theorem is not justified solely by those citations, nor does any step rename a fitted parameter as a prediction or smuggle an ansatz via self-reference. The argument structure remains independent of the target conclusion.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities can be extracted or verified from the given text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Totally nonnegative matrices, chain enumeration and zeros of polynomials." pith.science (2026). https://pith.science/paper/GANWTQC5

@misc{pith2026241206595,
  author       = {Pith},
  title        = {Pith review of: Totally nonnegative matrices, chain enumeration and zeros of polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GANWTQC5}},
  note         = {Machine review of arXiv:2412.06595}
}
abstract

We prove that any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for chain enumeration in posets and zeros of chain polynomials. The results obtained extend and unify results of the first author, Brenti, Welker and Athanasiadis. In the process we define a notion of $h$-vectors for a large class of posets which generalize the notions of $h$-vectors associated to simplicial and cubical complexes. A consequence of our methods is a characterization of the convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension and over a fixed finite field. This may be seen as a refinement of the Critical Problem of Crapo and Rota. We also use the methods developed to answer an open problem posed by Forg\'acs and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions.

Figures

Figures reproduced from arXiv: 2412.06595 by the authors.

Figure 1
Figure 1. The directed graph ΓN with its weights. We prove the existence of ΓN , N ∈ N, and λ by induction on N. Let R = (rn,k) N n,k=0 be a lower triangular matrix with nonnegative entries and with all di￾agonal entries equal to one, and let m + 1 be the first index for which rm+1,0 = 0. Whitney’s reduction theorem says that R is TN if and only if rj,0 = 0 for each j > m, and the matrix R˜ = (˜rn,k) N−1 n,k=0, where ˜rn,k = … view at source ↗
Figure 2
Figure 2. A quasi-rank uniform poset, and the corresponding quasi-rank function. We will primarily be interested in rank uniform posets, but find it more conve￾nient working in the more general setting of quasi-rank uniform posets. Let P be quasi-rank uniform. If x ∈ P, ρ(x) = n and 0 ≤ k ≤ n, let rn,k = rn,k(P) = |{z ∈ hxi : ρ(z) = k}|, and let R = R(P) = (rn,k) N n,k=0 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The Hasse diagram of Mr. Proposition 5.17. Let P be a binomial poset of rank N ∈ N ∪ {∞}, with factorial function B. Then P is TN if and only if the matrix (1/B(n−k))N n,k=0 is TN, where (1/B)(k) := 0 if k < 0. Hence if N = ∞, then P is TN if and only if {1/B(n)}∞ n=0is a Pólya frequency sequence. Proof. The numbers rn,k are given by rn,k = B(n) B(k)B(n − k) , see [18]. Since total nonnegativity is closed under mult… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The graphs of B7,4 and B4,3, respectively. Theorem 9.1. If 0 ≤ k ≤ n, then Rn,k(t) = σBn+1,k+1 (t)/t = S [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    R. M. Adin. A new cubical h-vector. In Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 199 4), volume 157, pages 3–14, 1996

    work page 1996

  2. [2]

    Aissen, A

    M. Aissen, A. Edrei, I. J. Schoenberg, and A. Whitney. On t he generating functions of totally positive sequences. Proc. Nat. Acad. Sci. U.S.A. , 37:303–307, 1951

    work page 1951

  3. [3]

    S. J. Alder. On q-simplicial posets . PhD thesis, University of East Anglia, 2010

    work page 2010

  4. [4]

    G. E. Andrews. The theory of partitions , volume Vol. 2 of Encyclopedia of Mathematics and its Applications. Addison-W esley Publishing Co., Reading, Mass.-London-A msterdam, 1976

    work page 1976

  5. [5]

    C. A. Athanasiadis. Face numbers of barycentric subdivi sions of cubical complexes. Israel J. Math., 246(1):423–439, 2021

    work page 2021

  6. [6]

    C. A. Athanasiadis, T. Douvropoulos, and K. Kalampogia- Evangelinou. Two classes of posets with real-rooted chain polynomials. Electron. J. Combin. , 31(4):Paper No. 4.16, 22, 2024

    work page 2024

  7. [7]

    C. A. Athanasiadis and K. Kalampogia-Evangelinou. Chai n enumeration, partition lattices and polynomials with only real roots. Comb. Theory , 3(1):Paper No. 12, 21, 2023

    work page 2023

  8. [8]

    Björner, A

    A. Björner, A. M. Garsia, and R. P. Stanley. An introducti on to Cohen-Macaulay partially ordered sets. In Ordered sets (Banff, Alta., 1981) , volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. , pages 583–615. Reidel, Dordrecht-Boston, Mass., 1982

    work page 1981

Show all 33 references
  1. [9]

    Borcea and P

    J. Borcea and P. Brändén. Multivariate Pólya-Schur clas sification problems in the Weyl algebra. Proc. Lond. Math. Soc. (3) , 101(1):73–104, 2010

    2010

  2. [10]

    P. Brändén. On linear transformations preserving the P ólya frequency property. Trans. Amer. Math. Soc. , 358(8):3697–3716, 2006

    2006

  3. [11]

    P. Brändén. Unimodality, log-concavity, real-rooted ness and beyond. In Handbook of enumer- ative combinatorics , Discrete Math. Appl. (Boca Raton), pages 437–483. CRC Pres s, Boca Raton, FL, 2015

    2015

  4. [12]

    Brändén and L

    P. Brändén and L. Saud Maia Leite. On chain polynomials o f geometric lattices, in prepara- tion

  5. [13]

    Brenti and V

    F. Brenti and V. W elker. f -vectors of barycentric subdivisions. Math. Z. , 259(4):849–865, 2008

    2008

  6. [14]

    H. H. Crapo and G.-C. Rota. On the foundations of combinatorial theory: Combinatorial geometries. MIT press Cambridge, Mass., 1970

    1970

  7. [15]

    Doubilet, G.-C

    P. Doubilet, G.-C. Rota, and R. Stanley. On the foundati ons of combinatorial theory. VI. The idea of generating function. In Proceedings of the Sixth Berkeley Symposium on Mathematica l Statistics and Probability (Univ. California, Berkeley, C alif., 1970/1971), Vol. II: Prob...

    1970

  8. [16]

    Ehrenborg and G

    R. Ehrenborg and G. Hetyei. Flags and shellings of Euler ian cubical posets. Ann. Comb. , 4(2):199–226, 2000

    2000

  9. [17]

    Ehrenborg and M

    R. Ehrenborg and M. Readdy. The r-cubical lattice and a generalization of the cd-index. European J. Combin. , 17(8):709–725, 1996

    1996

  10. [18]

    Ehrenborg and M

    R. Ehrenborg and M. A. Readdy. Sheffer posets and r-signed permutations. Ann. Sci. Math. Québec, 19(2):173–196, 1995

    1995

  11. [19]

    Forgács and K

    T. Forgács and K. Tran. Polynomials with rational gener ating functions and real zeros. J. Math. Anal. Appl. , 443(2):631–651, 2016

    2016

  12. [20]

    Z. Fu, Y. Peng, and Y. Zhang. The monoid representation o f upho posets and total positivity. arXiv preprint, arXiv:2411.04123 , 2024. TN-MATRICES, CHAIN ENUMERATION AND ZEROS OF POLYNOMIALS 31

    2024

  13. [21]

    S. R. Ghorpade, R. Pratihar, and T. H. Randrianarisoa. S hellability and homology of q- complexes and q-matroids. J. Algebraic Combin. , 56(4):1135–1162, 2022

    2022

  14. [22]

    S. Hopkins. Upho lattices I: Examples and non-examples of cores. arXiv preprint, arXiv:2407.08013, 2024

    2024

  15. [23]

    Jurrius and R

    R. Jurrius and R. Pellikaan. Defining the q-analogue of a matroid. Electron. J. Combin. , 25(3):Paper No. 3.2, 32, 2018

    2018

  16. [24]

    J. Neggers. Representations of finite partially ordere d sets. J. Combin. Inform. System Sci. , 3(3):113–133, 1978

    1978

  17. [25]

    V. Reiner. Upper binomial posets and signed permutatio n statistics. European J. Combin. , 14(6):581–588, 1993

    1993

  18. [26]

    G.-C. Rota. On the combinatorics of the Euler character istic. In Studies in Pure Mathematics (Presented to Richard Rado) , pages 221–233. Academic Press, London-New York, 1971

    1971

  19. [27]

    G. Royle. Personal communication, 2024

    2024

  20. [28]

    R. P. Stanley. Binomial posets, Möbius inversion, and p ermutation enumeration. J. Combi- natorial Theory Ser. A , 20(3):336–356, 1976

    1976

  21. [29]

    R. P. Stanley. Enumerative combinatorics. Volume 1 , volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2 012

  22. [30]

    R. P. Stanley. From Stern’s triangle to upper homogeneo us posets, 2020. Talk, transparencies, available at https://math.mit.edu/~rstan/transparencies/stern-ml.pdf

    2020

  23. [31]

    R. P. Stanley. Theorems and conjectures on some rationa l generating functions. European J. Combin., 119:Paper No. 103814, 18, 2024

    2024

  24. [32]

    J. R. Stembridge. Counterexamples to the poset conject ures of Neggers, Stanley, and Stem- bridge. Trans. Amer. Math. Soc. , 359(3):1115–1128, 2007

    2007

  25. [33]

    A. M. Whitney. A reduction theorem for totally positive matrices. J. Analyse Math. , 2:88–92, 1952. Department of Mathematics, KTH Royal Institute of Technolo gy, SE-100 44 Stock- holm, Sweden Email address : pbranden@kth.se, lsml@kth.se

    1952

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.