arXiv subjects

Petter Brändén

Publications and source records attributed to Petter Brändén.

At least 19 recordsLinked to original sources

arXiv2025

On chain polynomials of geometric lattices

Athanasiadis and Kalampogia-Evangelinou recently conjectured that the chain polynomial of any geometric lattice has only real zeros. We verify this conjecture for families of geometric lattices including perfect matroid designs, Dowling lattices, and for a class of geometric lattices that contains all lattices of flats of paving matroids. We also investigate how the conjecture behaves with respect to certain operations such as direct products, ordinal sums and single-element extensions.

math.CO↗

arXiv2025

Totally nonnegative matrices, chain enumeration and zeros of polynomials

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 Forgács and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions.

math.CO↗

arXiv2025

Lorentzian polynomials on cones

Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan. Several different characterizations of Lorentzian polynomials on cones are provided.

math.CO↗

arXiv2025

Preservation of inequalities under Hadamard products

Wagner (1992) proved that the Hadamard product of two Pólya frequency sequences that are interpolated by polynomials is again a Pólya frequency sequence. We study whether related combinatorial properties are preserved under Hadamard products. In particular, we show that ultra log-concavity, $γ$-positivity, and interlacing symmetric decompositions are preserved. Furthermore, we disprove a conjecture by Fischer and Kubitzke (2014) concerning the real-rootedness of Hadamard powers.

math.CO↗

arXiv2025

Chow polynomials of totally nonnegative matrices and posets

Huh-Stevens and Ferroni-Schröter independently conjectured that Hilbert-Poincaré series of Chow rings of geometric lattices have only real zeros. Ferroni, Matherne and the second author extended this conjecture to Chow polynomials of Cohen-Macaulay poset. In this paper we address the above conjectures by providing new defining relations and properties of Chow functions of posets and matrices. These are used, in conjunction with new techniques on interlacing sequences of polynomials, to prove that Chow polynomials of totally nonnegative matrices have only real zeros, which, in turn, proves the above conjectures for a class of posets that contains projective and affine geometries, face lattices of cubical polytopes, partition lattices and Dowling lattices, perfect matroid designs, and lattices of flats of paving matroids. We also study Chow polynomials of Toeplitz matrices in greater detail, and show how these are related the combinatorics of binomial and Sheffer posets, as well as to a family of generalized Eulerian polynomials with coefficients in the ring of symmetric polynomials that have been studied by e.g. Stanley, Brenti, Stembridge and Shareshian-Wachs.

math.CO↗

arXiv2025

Chow polynomials of uniform matroids are real-rooted

June Huh and Matthew Stevens conjectured that the Hilbert-Poincaré series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow polynomial and the augmented Chow polynomial of any maximally ranked poset has only real zeros.

math.CO↗

arXiv2024

Lorentzian polynomials

We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of Hodge--Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove that the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter q satisfies $0 < q \le 1$. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave sequence.

math.CO↗

arXiv2024

An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators

Given a linear ordinary differential operator T with polynomial coefficients, we study the class of closed subsets of the complex plane such that T sends any polynomial (resp. any polynomial of degree exceeding a given positive integer) with all roots in a given subset to a polynomial with all roots in the same subset or to 0. Below we discuss some general properties of such invariant subsets as well as the problem of existence of the minimal under inclusion invariant subset.

math.CA↗

arXiv2021

Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture

We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann relations of degree one for the Chow ring of a matroid.

math.CO↗

arXiv2021

The Eulerian transformation

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation $\mathcal{A} : \mathbb{R}[t] \to \mathbb{R}[t]$ defined by $\mathcal{A}(t^n) = A_n(t)$, where $A_n(t)$ denotes the $n$-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator $\mathcal{A}$, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.

math.CO↗

arXiv2021

Spaces of Lorentzian and real stable polynomials are Euclidean balls

We prove that projective spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls. This solves a conjecture of June Huh and the author. The proof utilizes and refines a connection between the symmetric exclusion process in Interacting Particle Systems and the geometry of polynomials.

math.CO↗

arXiv2021

Lower bounds for contingency tables via Lorentzian polynomials

We present a new lower bound on the number of contingency tables, improving upon and extending previous lower bounds by Barvinok and Gurvits. As an application, we obtain new lower bounds on the volumes of flow and transportation polytopes. Our proofs are based on recent results on Lorentzian polynomials.

math.CO↗

arXiv2020

Symmetric decompositions and real-rootedness

In algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternatingly increasing results pertaining to colored Eulerian and derangement polynomials, Ehrhart $h^\ast$-polynomials for lattice zonotopes, $h$-polynomials of barycentric subdivisions of doubly Cohen-Macaulay level simplicial complexes, and certain local $h$-polynomials for subdivisions of simplices. In particular, we prove two conjectures of Athanasiadis.

math.CO↗

arXiv2019

Some Algebraic Properties of Lecture Hall Polytopes

In this note, we investigate some of the fundamental algebraic and geometric properties of $s$-lecture hall simplices and their generalizations. We show that all $s$-lecture hall order polytopes, which simultaneously generalize $s$-lecture hall simplices and order polytopes, satisfy a property which implies the integer decomposition property. This answers one conjecture of Hibi, Olsen and Tsuchiya. By relating $s$-lecture hall polytopes to alcoved polytopes, we then use this property to show that families of $s$-lecture hall simplices admit a quadratic Gröbner basis with a square-free initial ideal. Consequently, we find that all $s$-lecture hall simplices for which the first order difference sequence of $s$ is a $0,1$-sequence have a regular and unimodular triangulation. This answers a second conjecture of Hibi, Olsen and Tsuchiya, and it gives a partial answer to a conjecture of Beck, Braun, Köppe, Savage and Zafeirakopoulos.

math.CO↗

arXiv2019

Hodge-Riemann relations for Potts model partition functions

We prove that the Hessians of nonzero partial derivatives of the (homogenous) multivariate Tutte polynomial of any matroid have exactly one positive eigenvalue on the positive orthant when $0<q\leq 1$. Consequences are proofs of the strongest conjecture of Mason and negative dependence properties for $q$-state Potts model partition functions.

math.CO↗

arXiv2018

Hyperbolic polynomials and the Kadison-Singer problem

Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison-Singer problem via Weaver's $KS_r$ conjecture. We extend this theorem to the realm of hyperbolic polynomials and hyperbolicity cones, as well as to arbitrary ranks. We also sharpen the theorem by providing better bounds, which imply better bounds in Weaver's $KS_r$ conjecture for each $r>2$. For $r=2$ our bound agrees with Bownik et al.

math.CO↗

arXiv2016

Lecture hall P-partitions

We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P,s)-Eulerian polynomials are real-rooted for certain pairs (P,s), and speculate on unimodality properties of these polynomials.

math.CO↗

arXiv2016

Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip into real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a "strong universal factor" as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.

math.CV↗