Improve this page

#Introduction

For an enumerative family of polynomials, natural questions include unimodality, real-rootedness, stability, and interlacing roots.

A useful general reference is [Br15], together with other works of P. Brändén. See also the IPAC seminar lecture on Lorentzian polynomials [Sem22].

The following pages cover more specialized topics:

#Unimodality, log-concavity and real roots

Let \(P = a_0 + a_1t+\dotsb + a_n t^n\) be a polynomial. It is called unimodal if there is some \(j\) such that \[a_0 \leq a_1 \leq \dotsb \leq a_j \geq a_{j+1} \geq a_{j+2} \geq \dotsb \geq a_n.\]

Suppose now all coefficients are positive. We say that \(P\) is log-concave if \[a_{j-1}a_{j+1} \leq a_j^2\] for all relevant \(j.\) This property implies unimodality.

We get an even stronger property if the Newton inequalities are satisfied: \[\frac{a_{j-1}}{ \binom{n}{j-1} } \frac{a_{j+1}}{ \binom{n}{j+1} } \leq \frac{a_{j}^2}{ \binom{n}{j}^2 }.\]

Finally, if \(P\) has only real roots, it is called real-rooted. Every real-rooted polynomial satisfies the Newton inequalities, see e.g. [Br15].

In [Wan02], a conjecture by R. Simion is proved: a sequence of lattice path counts is log-concave. See also [Sag87] for a short proof of unimodality in this case.

In [AB18] it is shown that if \(f(t)\) has non-negative, non-decreasing coefficients, then the polynomial \(f(1+x+x^2+\dotsb+x^\ell)\) is unimodal, for any \(\ell \geq 1.\) E. Y. H. Li, G. M. X. Li, A. L. B. Yang, and Z.-X. Zhang use chromatic symmetric functions to prove log-concavity results for independence polynomials [LLYZ25]. Their applications include spiders and pineapple graphs. J. Liang and B. E. Sagan give a broad distributive lattice method for proving log-concavity and log-convexity [LS24]. Their Order Ideal Lemma applies to lattice paths, Young’s lattice intervals, order polynomials, Schur and Schur \(Q\)-function specializations, Lucas sequences, descent and peak polynomials, pattern avoidance, set partitions, and noncrossing partitions.

#Classical criteria and root counts

Several classical criteria extract root information directly from the coefficients. Descartes’ rule bounds roots of each sign, the Hermite–Sylvester theorem gives an exact semidefinite certificate for real-rootedness, and Sturm’s theorem counts roots in any interval.

For a real polynomial \[f(t)=a_nt^n+a_{n-1}t^{n-1}+\dotsb+a_0,\] let \(V(f)\) be the number of sign changes in \((a_n,a_{n-1},\dotsc,a_0)\) after the zero entries are omitted.

Theorem (Descartes’ rule of signs).

Let \(N_+(f)\) be the number of positive real zeros of \(f,\) counted with multiplicity. Then \[V(f)-N_+(f)\] is a non-negative even integer. The corresponding statement for negative zeros is obtained by applying the rule to \(f(-t).\)

For the Hermite–Sylvester criterion, normalize \(f\) to be monic and write \[f(t)=t^n+c_1t^{n-1}+\dotsb+c_n =\prod_{j=1}^n(t-r_j),\] where the roots are repeated according to multiplicity. Define the Newton sums \[s_k(f)=\sum_{j=1}^n r_j^k, \qquad s_0(f)=n.\] These power sums do not require knowing the roots. The Newton identities express them recursively in the coefficients: \[s_k+c_1s_{k-1}+\dotsb+c_{k-1}s_1+kc_k=0 \qquad (1\leq k\leq n),\] and \[s_k+c_1s_{k-1}+\dotsb+c_ns_{k-n}=0 \qquad (k\gt{}n).\]

Theorem (Hermite–Sylvester, [Nat19]).

Let \(f\in\setR[t]\) be monic of degree \(n,\) and form its Hermite matrix \[H_f=\bigl(s_{i+j}(f)\bigr)_{0\leq i,j\leq n-1}.\] Then \(f\) is real-rooted if and only if \(H_f\) is positive semidefinite.

Thus Hermite–Sylvester is a criterion purely in the coefficients of \(f:\) Newton’s identities determine every entry of \(H_f\) up to \(s_{2n-2}.\)

#Kurtz theorem

Theorem (Kurtz sufficiency theorem, [Hut23, Kur92]).

D. Kurtz proved that a polynomial \(a_n t^n + a_{n-1}t^{n-1} + \dotsb + a_1 t + a_0\) with positive coefficients (and degree at least two) that satisfies \[a_i^2 \gt 4a_{i-1}a_{i+1} \text{ for all } i,\] is real-rooted.

This result was proved earlier in the context of entire functions, see [Hut23].

#Transformations preserving real-rootedness

Several standard methods prove real-rootedness by applying a transformation known to preserve real roots. Coefficientwise methods based on Pólya frequency sequences and totally non-negative matrices are collected on their own page; the main examples there include Aissen–Schoenberg–Whitney, Edrei–Thoma theory, Veronese sections, and path matrices. This page records two other frequently used classes of transformations: basis transformations such as the Narayana and Eulerian transformations, and coefficientwise products such as Hadamard products.

#Narayana transformations and matrix products

J. Mao and L. Wang study a basis transformation interpolating between the type \(B\) and type \(A\) Narayana polynomials [MW26]. For \(m\in \setZ_{\geq 0},\) set \[N_{n,m}(x) \coloneqq {}_2F_1(-n,-n-m;m+1;x) = \sum_{k=0}^n \frac{(-n)_k(-n-m)_k}{(m+1)_k\,k!}x^k,\] where \((a)_k\) is the Pochhammer symbol. Then \(N_{n,0}(x)\) is the type \(B\) Narayana polynomial and \(N_{n,1}(x)\) is the type \(A\) Narayana polynomial. The corresponding Narayana transformation \(T_{N_m}\) is defined by \[T_{N_m}(x^n)=N_{n,m}(x).\]

Theorem (Mao–Wang, [Thm. 1.1, MW26]).

Let \(m\in\setZ_{\geq 0}.\) Suppose that \[p(x)=\sum_{k=0}^n a_kx^k \in \setR_{\geq 0}[x]\] has only real roots and has degree \(n.\) Then \[T_{N_m}(p(x))=\sum_{k=0}^n a_kN_{k,m}(x)\] has only real nonpositive roots.

In particular, the type \(A\) and type \(B\) Narayana transformations preserve real-rootedness for polynomials with non-negative coefficients. The proof uses rectangular additive convolution, and the non-negativity assumption on the coefficients is essential. This should be compared with the Eulerian transformation \(A(x^n)=A_n(x).\) Brenti conjectured that the Eulerian transformation preserves real-rootedness, but this is false in full generality by P. Brändén and K. Jochemko [BJ22]. A positive result in this direction is due to C. Athanasiadis, who proves real-rootedness for a large non-negative cone; see the gamma-positivity page and [Ath25].

Theorem (Brenti, [Thm. 2.4.2, Bre95]).

Let \[\langle x\rangle_k\coloneqq x(x-1)\dotsm(x-k+1), \qquad \langle x\rangle_0\coloneqq 1.\] If \[f(x)=\sum_{k=0}^n a_k\langle x\rangle_k\] has only real nonpositive roots, then \[\sum_{k=0}^n a_kx^k\] has only real nonpositive roots.

Mao and Wang use this theorem as Lemma 3.2 of [MW26] in their study of products of lower triangular matrices. If \(M=(M(n,k))_{n,k\geq 0}\) is lower triangular, write \[M_n(x)\coloneqq \sum_{k=0}^n M(n,k)x^k\] for its row-generating function (RGF). Their product criterion gives a broad class of matrices \(B\) for which real-rootedness of all \(M_n(x)\) forces real-rootedness of the RGFs of \(MB^r\) for every \(r\geq 1\) [Thm. 1.3, MW26]. This applies when \(B\) is the coefficient matrix of the polynomials \(N_{n,m}(x),\) and also to recurrence-defined triangular matrices including Pascal’s triangle and the Stirling triangles.

Conjecture (Mao–Wang, [Conj. 4.1, MW26]).

Let \(A\) be the Eulerian triangle and let \(D\) be the Delannoy triangle. The row-generating functions of \(A^2\) and \(D^2\) have only real nonpositive roots.

#Coefficientwise products and multiplier sequences

Coefficientwise products are another standard way to preserve real roots. There are three closely related forms in common use: finite Hadamard product theorems, infinite diagonal operators or multiplier sequences, and normalized products such as the Schur–Szegő composition and diamond product.

The Hadamard product (or coefficientwise product) of two polynomials \(f(t)=\sum a_j t^j\) and \(g(t)=\sum b_j t^j\) is \[f \odot g \coloneqq \sum_j a_j b_j \, t^j.\]

#Multiplier sequences

Definition (Multiplier sequence).

A sequence \(\gamma=(\gamma_0,\gamma_1,\dotsc)\) is a multiplier sequence if the diagonal operator \[T_\gamma\!\left(\sum_{k\geq 0} a_k x^k\right) \coloneqq \sum_{k\geq 0} \gamma_k a_k x^k\] sends every real-rooted polynomial to a real-rooted polynomial or to zero.

Theorem (Pólya–Schur, see [SP14, Kar68]).

A real sequence \(\gamma=(\gamma_0,\gamma_1,\dotsc)\) is a multiplier sequence if and only if the exponential generating function \[\Phi_\gamma(x)=\sum_{k\geq 0}\gamma_k\frac{x^k}{k!}\] or \(\Phi_\gamma(-x)\) has the form \[Cx^m e^{ax}\prod_{j\geq 1}(1+\alpha_jx),\] where \(C\in\setR,\) \(m\geq 0,\) \(a\geq 0,\) \(\alpha_j\geq 0,\) and \(\sum_j\alpha_j\lt{}\infty.\)

There is also a finite-degree test, which is often easier to use than the entire-function classification.

Theorem (Finite multiplier-sequence criterion, [Thm. 3.14, Br06]).

Fix \(n\geq 0.\) The diagonal operator \[T_\gamma(x^k)=\gamma_kx^k, \qquad 0\leq k\leq n,\] preserves real-rootedness on polynomials of degree at most \(n\) if and only if \[\sum_{k=0}^n \binom{n}{k}\gamma_kx^k\] is zero or has only real zeros, all in one of the two half-lines \((-\infty,0]\) and \([0,\infty).\)

#Hadamard product theorems

Theorem (Schur–Pólya; see [Thm. 3.4, Wag92]).

If \(f\) and \(g\) are real-rooted with non-negative coefficients, then \(f\odot g\) is also real-rooted with non-negative coefficients.

Theorem (Garloff–Wagner, see [Thm. 4, GW96]).

Let \(f,g,p,q \in \setR[t]\) have non-negative coefficients and only real non-positive zeros. If \[f \interl g \qquad\text{and}\qquad p \interl q,\] then \[f\odot p \interl g\odot q.\] Here we use the zero convention for interleaving, so the conclusion also covers the case where one of the Hadamard products is zero.

Theorem (Schur, see [Sch14] and [Thm. 1, Br04]).

Let \[f(x)=a_0+a_1x+\dotsb+a_nx^n, \qquad g(x)=b_0+b_1x+\dotsb+b_mx^m\] be real-rooted polynomials. Suppose that all zeros of \(g\) have the same sign. Then \[\sum_{k\geq 0} k!\,a_kb_kx^k\] is real-rooted, where missing coefficients are interpreted as zero. If \(a_0b_0\neq 0,\) then all zeros are distinct.

J. S. Kim and J. Oh study Hadamard products of Jacobi–Trudi matrices in relation to total positivity [KO25]. They give Schur-positive results for ribbon-indexed Hadamard squares; the arXiv version notes that an attempted bijection has a flaw, so Sokal’s conjecture on Hadamard squares remains open.

Theorem (Laguerre, see [Thm. 12, Br04]).

If \[f(x)=a_0+a_1x+a_2x^2+\dotsb+a_nx^n\] is real-rooted, then \[a_0+a_1x+\frac{a_2}{2!}x^2+\dotsb+\frac{a_n}{n!}x^n\] is real-rooted.

The Schur–Pólya theorem follows from the fact that the class of polynomials with only real non-positive zeros and non-negative coefficients is a cone of multiplier sequences, closed under the Hadamard product. See [Kar68] or the exposition in [Br15].

Example (Type \(B\) Narayana polynomials via Hadamard product).

The type \(B\) Narayana polynomials \(\sum_{k=0}^n \binom{n}{k}^2 t^k\) are the Hadamard product of \((1+t)^n\) with itself. Since \((1+t)^n\) is real-rooted, the Hadamard product theorem immediately gives real-rootedness.

The Hadamard product theorem is useful whenever a polynomial’s coefficients factor as a product of two sequences that are each the coefficient sequence of a real-rooted polynomial.

#Schur–Szegő composition

Theorem (Schur–Szegő composition theorem, see [RS02]).

Let \[F(x)=\sum_{k=0}^n \binom{n}{k} a_k x^k \qquad\text{and}\qquad G(x)=\sum_{k=0}^n \binom{n}{k} b_k x^k.\] Their Schur–Szegő composition is \[(F*G)(x) \coloneqq \sum_{k=0}^n \binom{n}{k} a_k b_k x^k.\] If \(F\) and \(G\) have only real non-positive zeros, then \(F*G\) also has only real non-positive zeros.

#Diamond product

Theorem (Wagner–Brändén, [Thm. 2 and Thm. 13, Br04]).

For polynomials \(f,h\in \setR[x],\) define the diamond product by \[f\diamond h \coloneqq \sum_{n\geq 0} \frac{f^{(n)}(x)h^{(n)}(x)}{n!\,n!}x^n(x+1)^n.\] If \(h\) has all its zeros in the interval \([-1,0]\) and \(f\) is real-rooted, then \(f\diamond h\) is real-rooted. Moreover, if \(g \interl f,\) then \[g\diamond h \interl f\diamond h.\] In the special case where both \(f\) and \(h\) have all zeros in \([-1,0],\) the polynomial \(f\diamond h\) also has all zeros in \([-1,0].\)

#Poset operations preserving real-rootedness

#Neggers–Stanley conjecture

For a naturally labeled poset \(P,\) the \(P\)-Eulerian polynomial is the descent generating polynomial of the linear extensions of \(P.\) The Neggers–Stanley conjecture predicted that these polynomials have only real zeros. The conjecture is false in this generality: counterexamples were constructed by J. R. Stembridge [Ste07]. Positive results for restricted classes of posets are still important, especially because \(P\)-Eulerian polynomials are \(h^*\)-polynomials of order polytopes.

Example (P-Eulerian polynomials for series-parallel posets).

Let \(E(P,t)\) denote the Eulerian polynomial associated with the naturally labeled poset \(P.\) The ordinal sum \(P \oplus Q\) of two posets is the poset where \[x \prec y \iff x \prec_P y \text{ or } x \prec_Q y \text{ or } x \in P, y \in Q.\] The Eulerian polynomial then has the property that \[E(P \oplus Q, t) = E(P,t) \cdot E(Q,t).\] Moreover, for the disjoint union of posets, we have \[E(P \sqcup Q, t) = E(P,t) \diamond E(Q,t),\] where \(\diamond\) is the diamond product above. These operations preserve real-rootedness, so all series-parallel posets have real-rooted Eulerian polynomials, see [Wag92, Wag92]. This result extends to labeled posets, see [Br04].

#Generalized snake posets

B. Braun and A. Jal prove \(h^*\)-real-rootedness for order polytopes of generalized snake posets [BJ26]. If \(w\) is a generalized snake word and \(P(w)\) is the corresponding naturally labeled width-two poset, then Stanley’s theorem and the width-two poset model of P. Alexandersson and A. Jal identify \[h^*(\mathcal{O}(P(w));t) = W_{P(w)}(t) = M_w(t),\] where \(W_{P(w)}(t)\) is the descent-generating polynomial of the linear extensions of \(P(w),\) and \(M_w(t)\) is the corresponding non-nesting rook polynomial.

Theorem (Braun–Jal [Thm. 4.1, BJ26]).

Let \(w\) be a word in \(L\) and \(R\) of positive length, and let \(w'\) be obtained by deleting the last letter of \(w.\) Then \(M_{\epsilon w}(t)\) is real-rooted and \[M_{\epsilon w'}(t) \interl M_{\epsilon w}(t).\]

The proof is a clean interlacing induction using a recurrence for the non-nesting rook polynomials. The constant words recover the Narayana polynomials, while the regular snake words recover the Delannoy row polynomials \(P_n(t),\) whose coefficients are the antidiagonal rows of A008288. These polynomials satisfy \[P_0(t)=1,\qquad P_1(t)=1+t,\qquad P_n(t)=(1+t)P_{n-1}(t)+tP_{n-2}(t) \quad (n\geq 2).\] Thus the Delannoy row polynomials are \(h^*\)-polynomials of order polytopes of regular snake posets. This is a different family from the zig-zag Eulerian polynomials \(Z_m(t)\) discussed on the gamma-positivity page; those polynomials are known to be gamma-positive, but their real-rootedness is a separate open problem. In particular, the first nontrivial regular-snake base case is \[M_{\epsilon R}(t)=M_{\epsilon L}(t)=1+3t+t^2.\] This is also the first nontrivial row \(1,3,1\) of the Delannoy triangle A008288, and the same polynomial appears in the weighted transfer-matrix tiling example.

#Normality in the limit

If \(P_n(x)\) is a sequence of real-rooted polynomials with non-negative coefficients, the coefficients often become asymptotically normally distributed. The following theorem makes this precise.

Theorem (Bender 1973, Harper 1967).

Suppose \[P_n(x) = \sum_{k} a_n(k) x^k = a_n \prod_j (x + r_j)\] where \(r_j \geq 0.\) Define \[\mu_n = \sum_j \frac{1}{1 + r_j}, \quad \sigma_n^2 = \sum_j \frac{r_j}{(1 + r_j)^2}.\] If \(\sigma_n \to \infty,\) then the normalized coefficients \[p_n(k) = \frac{a_n(k)}{P_n(1)}\] are asymptotically normal: if \(X_n\) is the random variable with distribution \(p_n(k),\) then \[\frac{X_n - \mu_n}{\sigma_n} \xrightarrow{d} \mathcal{N}(0,1).\] Moreover, if the \(a_n(k)\) are log-concave (which follows from real-rootedness), a local limit theorem holds: \[p_n(k) \approx \frac{1}{\sigma_n \sqrt{2 \pi}} e^{ - (k - \mu_n)^2 / (2 \sigma_n^2) }\] uniformly in \(k.\)

See [Ben73] for the original result and examples including ordered set partitions, Eulerian numbers, and matchings on the \(2\times n\) grid graph. For Baxter permutations, see [Zha24]. For P-recursively defined polynomials, see [Li25]. Several further examples are provided in [Hit24].

Bibliography

  1. [AB18]Moussa Ahmia and Hacène Belbachir. Unimodality polynomials and generalized Pascal triangles. Algebra and Discrete Mathematics, 26(1):1–7, 2018.
    .bib
    @article{AhmiaBelbachir2018,
  author    = {Ahmia, Moussa and Belbachir, Hac{\`e}ne},
  title     = {Unimodality polynomials and generalized {P}ascal triangles},
  journal   = {Algebra and Discrete Mathematics},
  volume    = {26},
  number    = {1},
  pages     = {1--7},
  year      = {2018},
  issn      = {1726-3255},
  url       = {https://admjournal.luguniv.edu.ua/index.php/adm/article/view/193}
}
  2. [Ath25]Christos A. Athanasiadis. On the real-rootedness of the Eulerian transformation. Journal of the London Mathematical Society, 111(2), 2025.
    .bib
    @article{Athanasiadis2025EulerianTransformation,
  author = {Athanasiadis, Christos A.},
  title = {On the real-rootedness of the {E}ulerian transformation},
  journal = {Journal of the London Mathematical Society},
  volume = {111},
  number = {2},
  year = {2025},
  doi = {10.1112/jlms.70083},
  url2 = {https://doi.org/10.1112/jlms.70083},
  eprint = {2302.00754},
  archivePrefix = {arXiv}
}
  3. [Ben73]Edward A Bender. Central and local limit theorems applied to asymptotic enumeration. Journal of Combinatorial Theory, Series A, 15(1):91–111, July 1973.
    .bib
    @article{Bender1973,
  title = {Central and local limit theorems applied to asymptotic enumeration},
  volume = {15},
  ISSN = {0097-3165},
  url = {http://dx.doi.org/10.1016/0097-3165(73)90038-1},
  DOI = {10.1016/0097-3165(73)90038-1},
  number = {1},
  journal = {Journal of Combinatorial Theory,  Series A},
  publisher = {Elsevier BV},
  author = {Bender,  Edward A},
  year = {1973},
  month = jul,
  pages = {91–-111}
}
  4. [Br04]Petter Brändén. On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture. Journal of Algebraic Combinatorics, 20(2):119–130, September 2004.
    .bib
    @article{Branden2004operators,
  title = {On Operators on Polynomials Preserving Real-Rootedness and the {N}eggers-{S}tanley Conjecture},
  volume = {20},
  ISSN = {0925-9899},
  url2 = {http://dx.doi.org/10.1023/B:JACO.0000047295.93525.df},
  DOI = {10.1023/b:jaco.0000047295.93525.df},
  number = {2},
  journal = {Journal of Algebraic Combinatorics},
  publisher = {Springer Science and Business Media LLC},
  author = {Petter Br{\"{a}}nd{\'{e}}n},
  year = {2004},
  month = sep,
  pages = {119–130}
}
  5. [Br06]Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006.
    .bib
    @article{Branden2006,
  title = {On linear transformations preserving the {P}{\'{o}}lya frequency property},
  volume = {358},
  ISSN = {1088-6850},
  url = {http://dx.doi.org/10.1090/S0002-9947-06-03856-6},
  DOI = {10.1090/s0002-9947-06-03856-6},
  number = {8},
  journal = {Transactions of the American Mathematical Society},
  publisher = {American Mathematical Society (AMS)},
  author = {Br{\"a}nd{\'e}n, Petter},
  year = {2006},
  month = feb,
  pages = {3697--3716}
}
  6. [Br15]Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics:437–483, March 2015.
    .bib
    @incollection{Branden2015,
  doi = {10.1201/b18255-10},
  year = {2015},
  month = mar,
  publisher = {Chapman and Hall/{CRC}},
  pages = {437--483},
  author = {Petter Br{\"{a}}nd{\'{e}}n},
  title = {Unimodality, Log-Concavity, Real-Rootedness and Beyond},
  booktitle = {Handbook of Enumerative Combinatorics}
}
  7. [BJ22]Petter Brändén and Katharina Jochemko. The Eulerian transformation. Transactions of the American Mathematical Society, 375(3):1917–1931, 2022.
    .bib
    @article{BrandenJochemko2022,
  author = {Petter Br{\"a}nd{\'e}n and Katharina Jochemko},
  title = {The {E}ulerian transformation},
  journal = {Transactions of the American Mathematical Society},
  volume = {375},
  number = {3},
  pages = {1917--1931},
  year = {2022},
  doi = {10.1090/tran/8539},
  eprint = {2103.00890},
  url = {https://arxiv.org/abs/2103.00890}
}
  8. [BJ26]Benjamin Braun and Aryaman Jal. Order polytopes of generalized snake posets are $h^*$-real-rooted. arXiv:2607.00922, 2026.
    .bib
    @article{BraunJal2026x,
  author = {Benjamin Braun and Aryaman Jal},
  title = {Order polytopes of generalized snake posets are {$h^*$}-real-rooted},
  year = {2026},
  eprint = {2607.00922},
  url = {https://arxiv.org/abs/2607.00922},
  journal = {arXiv e-prints}
}
  9. [Bre95]Francesco Brenti. Combinatorics and total positivity. Journal of Combinatorial Theory, Series A, 71(2):175–218, 1995.
    .bib
    @article{Brenti1995,
  author = {Brenti, Francesco},
  title = {Combinatorics and total positivity},
  year = {1995},
  journal = {Journal of Combinatorial Theory, Series A},
  volume = {71},
  number = {2},
  pages = {175--218},
  publisher = {Elsevier BV},
  doi = {10.1016/0097-3165(95)90000-4},
  url = {http://dx.doi.org/10.1016/0097-3165(95)90000-4},
  issn = {0097-3165}
}
  10. [GW96]Jürgen Garloff and David G. Wagner. Hadamard products of stable polynomials are stable. Journal of Mathematical Analysis and Applications, 202(3):797–809, 1996.
    .bib
    @article{GarloffWagner1996,
  doi = {10.1006/jmaa.1996.0348},
  url2 = {https://doi.org/10.1006/jmaa.1996.0348},
  year = {1996},
  publisher = {Elsevier {BV}},
  volume = {202},
  number = {3},
  pages = {797--809},
  author = {J{\"u}rgen Garloff and David G. Wagner},
  title = {Hadamard Products of Stable Polynomials Are Stable},
  journal = {Journal of Mathematical Analysis and Applications}
}
  11. [Hit24]Paweł Hitczenko. A class of polynomial recurrences resulting in $(n/\log n, n/\log^2n)$-asymptotic normality. arXiv:2403.03422, 2024.
    .bib
    @article{Hitczenko2024x,
  Author  = {Pawe{\l} Hitczenko},
  Title   = {A class of polynomial recurrences resulting in $(n/\log n, n/\log^2n)$-asymptotic normality},
  Year    = {2024},
  journal = {arXiv e-prints},
  Eprint  = {2403.03422}
}
  12. [Hut23]J. I. Hutchinson. On a remarkable class of entire functions. Transactions of the American Mathematical Society, 25(3):325–332, 1923.
    .bib
    @article{Hutchinson1923,
  author = {Hutchinson, J. I.},
  title = {On a remarkable class of entire functions},
  year = {1923},
  journal = {Transactions of the American Mathematical Society},
  volume = {25},
  number = {3},
  pages = {325–332},
  publisher = {American Mathematical Society (AMS)},
  doi = {10.1090/s0002-9947-1923-1501248-1},
  url = {http://dx.doi.org/10.1090/s0002-9947-1923-1501248-1},
  issn = {0002-9947}
}
  13. [Sem22]IPAC Seminar. Lorentzian polynomials lecture 1. Video lecture, 2022.
    .bib
    @misc{IPAC2022LorentzianLecture,
  author = {{IPAC Seminar}},
  title = {Lorentzian Polynomials Lecture 1},
  howpublished = {Video lecture},
  year = {2022},
  url = {https://www.youtube.com/watch?v=wuQN0xaTkxE}
}
  14. [Kar68]Samuel Karlin. Total positivity, Vol. I. Stanford University Press, 1968.
    .bib
    @book{Karlin1968,
  author    = {Karlin, Samuel},
  title     = {Total Positivity, {V}ol.~{I}},
  publisher = {Stanford University Press},
  year      = {1968}
}
  15. [KO25]Jang Soo Kim and Jaeseong Oh. Total positivity of Hadamard product of dual Jacobi–Trudi matrices. arXiv:2504.12583, 2025.
    .bib
    @article{KimOh2025x,
  author = {Jang Soo Kim and Jaeseong Oh},
  title = {Total positivity of {H}adamard product of dual {J}acobi--{T}rudi matrices},
  year = {2025},
  eprint = {2504.12583},
  url = {https://arxiv.org/abs/2504.12583},
  journal = {arXiv e-prints}
}
  16. [Kur92]David C. Kurtz. A sufficient condition for all the roots of a polynomial to be real. The American Mathematical Monthly, 99(3):259–263, 1992.
    .bib
    @article{Kurtz1992,
  author = {Kurtz, David C.},
  title = {A Sufficient Condition for All the Roots of a Polynomial To Be Real},
  year = {1992},
  journal = {The American Mathematical Monthly},
  volume = {99},
  number = {3},
  pages = {259--263},
  publisher = {Informa UK Limited},
  doi = {10.1080/00029890.1992.11995845},
  url = {http://dx.doi.org/10.1080/00029890.1992.11995845},
  issn = {1930-0972}
}
  17. [Li25]Zhongjie Li. Asymptotic normality of coefficients of P-recursive polynomial sequences. arXiv:2504.11865, 2025.
    .bib
    @article{Li2025x,
  Author  = {Zhongjie Li},
  Title   = {Asymptotic normality of coefficients of {P}-recursive polynomial sequences},
  Year    = {2025},
  journal = {arXiv e-prints},
  Eprint  = {2504.11865}
}
  18. [LLYZ25]Ethan Y. H. Li, Grace M. X. Li, Arthur L. B. Yang and Zhong-Xue Zhang. A symmetric function approach to log-concavity of independence polynomials. arXiv:2501.04245, 2025.
    .bib
    @article{LiLiYangZhang2025x,
  author = {Ethan Y. H. Li and Grace M. X. Li and Arthur L. B. Yang and Zhong-Xue Zhang},
  title = {A symmetric function approach to log-concavity of independence polynomials},
  year = {2025},
  eprint = {2501.04245},
  url = {https://arxiv.org/abs/2501.04245},
  journal = {arXiv e-prints}
}
  19. [LS24]Jinting Liang and Bruce E. Sagan. Log-concavity and log-convexity via distributive lattices. arXiv:2408.02782, 2024.
    .bib
    @article{LiangSagan2024x,
  author = {Jinting Liang and Bruce E. Sagan},
  title = {Log-concavity and log-convexity via distributive lattices},
  year = {2024},
  eprint = {2408.02782},
  url = {https://arxiv.org/abs/2408.02782},
  journal = {arXiv e-prints}
}
  20. [MW26]Jianxi Mao and Lijie Wang. The Narayana transformation. arXiv:2607.01572, 2026.
    .bib
    @article{MaoWang2026x,
  author = {Jianxi Mao and Lijie Wang},
  title = {The {N}arayana transformation},
  year = {2026},
  eprint = {2607.01572},
  url = {https://arxiv.org/abs/2607.01572},
  journal = {arXiv e-prints}
}
  21. [Nat19]Melvyn B. Nathanson. The Hermite-Sylvester criterion for real-rooted polynomials. arXiv:1911.01745v2, 2019.
    .bib
    @article{Nathanson2019x,
  author = {Melvyn B. Nathanson},
  title = {The {H}ermite-{S}ylvester criterion for real-rooted polynomials},
  year = {2019},
  eprint = {1911.01745v2},
  url = {https://arxiv.org/abs/1911.01745v2},
  journal = {arXiv e-prints},
  journalref = {The Mathematical Gazette 105 (2021), 122--125},
  doi = {10.1017/mag.2021.19}
}
  22. [RS02]Q. I. Rahman and G. Schmeisser. Analytic theory of polynomials. London mathematical society monographs. Oxford University Press, 2002.
    .bib
    @book{RahmanSchmeisser2002,
  author    = {Rahman, Q. I. and Schmeisser, G.},
  title     = {Analytic Theory of Polynomials},
  publisher = {Oxford University Press},
  series    = {London Mathematical Society Monographs},
  year      = {2002}
}
  23. [Sag87]Bruce E Sagan. Unimodality and the reflection principle. Ars Combinatoria, 24:27–40, 1987.
    .bib
    @article{Sagan1987unimodal,
  title={Unimodality and the reflection principle},
  author={Sagan, Bruce E},
  journal={Ars Combinatoria},
  volume={24},
  pages={27--40},
  year={1987}
}
  24. [Sch14]Issai Schur. Zwei sätze über algebraische Gleichungen mit lauter reellen Wurzeln. Journal für die reine und angewandte Mathematik, 144(2):75–88, 1914.
    .bib
    @article{Schur1914,
  title = {Zwei S{\"a}tze {\"u}ber algebraische {G}leichungen mit
           lauter reellen {W}urzeln},
  author = {Schur, Issai},
  year = {1914},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {144},
  number = {2},
  pages = {75--88},
  doi = {10.1515/crll.1914.144.75},
  url2 = {https://doi.org/10.1515/crll.1914.144.75},
  publisher = {Walter de Gruyter}
}
  25. [SP14]J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
    .bib
    @article{SchurPolya1914,
  title={{\"U}ber zwei {A}rten von {F}aktorenfolgen in der {T}heorie der algebraischen {G}leichungen},
  author={Schur, J and Polya, G},
  year={1914},
  journal={J. Reine Angew. Math.},
  publisher={Walter de Gruyter, Berlin/New York Berlin, New York}
}
  26. [Ste07]John R. Stembridge. Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge. Trans. Amer. Math. Soc., 359(3):1115–1128, 2007.
    .bib
    @article {Stembridge2007NeggersStanleyCounter,
    AUTHOR = {Stembridge, John R.},
     TITLE = {Counterexamples to the poset conjectures of {N}eggers,
              {S}tanley, and {S}tembridge},
   JOURNAL = {Trans. Amer. Math. Soc.},
  FJOURNAL = {Transactions of the American Mathematical Society},
    VOLUME = {359},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {1115--1128},
      ISSN = {0002-9947,1088-6850},
   MRCLASS = {06A07 (05A15)},
  MRNUMBER = {2262844},
MRREVIEWER = {David\ B.\ Penman},
       DOI = {10.1090/S0002-9947-06-04271-1},
       URL = {https://doi.org/10.1090/S0002-9947-06-04271-1},
}
  27. [Wag92]David G Wagner. Total positivity of Hadamard products. Journal of Mathematical Analysis and Applications, 163(2):459–483, January 1992.
    .bib
    @article{Wagner1992,
  doi = {10.1016/0022-247x(92)90261-b},
  url2 = {https://doi.org/10.1016/0022-247x(92)90261-b},
  year = {1992},
  month = jan,
  publisher = {Elsevier {BV}},
  volume = {163},
  number = {2},
  pages = {459--483},
  author = {David G Wagner},
  title = {Total positivity of {H}adamard products},
  journal = {Journal of Mathematical Analysis and Applications}
}
  28. [Wag92]David G. Wagner. Enumeration of functions from posets to chains. European Journal of Combinatorics, 13(4):313–324, July 1992.
    .bib
    @article{Wagner1992en,
  title = {Enumeration of functions from posets to chains},
  volume = {13},
  ISSN = {0195-6698},
  url = {http://dx.doi.org/10.1016/S0195-6698(05)80036-8},
  DOI = {10.1016/s0195-6698(05)80036-8},
  number = {4},
  journal = {European Journal of Combinatorics},
  publisher = {Elsevier BV},
  author = {Wagner,  David G.},
  year = {1992},
  month = jul,
  pages = {313–324}
}
  29. [Wan02]Yi Wang. A simple proof of a conjecture of Simion. Journal of Combinatorial Theory, Series A, 100(2):399–402, November 2002.
    .bib
    @article{Wang2002,
  title = {A Simple Proof of a Conjecture of {S}imion},
  volume = {100},
  ISSN = {0097-3165},
  url = {http://dx.doi.org/10.1006/jcta.2002.3300},
  DOI = {10.1006/jcta.2002.3300},
  number = {2},
  journal = {Journal of Combinatorial Theory,  Series A},
  publisher = {Elsevier BV},
  author = {Wang,  Yi},
  year = {2002},
  month = nov,
  pages = {399--402}
}
  30. [Zha24]James Jing Yu Zhao. Asymptotic normality arising in Baxter permutations. arXiv:2410.05031, 2024.
    .bib
    @article{Zhao2024x,
  Author  = {James Jing Yu Zhao},
  Title   = {Asymptotic normality arising in {B}axter permutations},
  Year    = {2024},
  journal = {arXiv e-prints},
  Eprint  = {2410.05031}
}

I use cookies to detect website issues and track search terms.