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Petter Brändén

Publications and source records attributed to Petter Brändén.

53 records · Page 3Linked to original sources
arXiv2010

On the half-plane property and the Tutte group of a matroid

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.

math.CO↗
arXiv2009

Multivariate Polya-Schur classification problems in the Weyl algebra

A multivariate polynomial is {\em stable} if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra $\A_n$ that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur's notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in $\A_1$ that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in $\A_n$ preserves stability if and only if its Fischer-Fock adjoint does. These are powerful multivariate extensions of the classical Hermite-Poulain-Jensen theorem, Pólya's curve theorem and Schur-Maló-Szegő composition theorems. Examples and applications to strict stability preservers are also discussed.

math.CA↗
arXiv2009

A generalization of the Heine--Stieltjes theorem

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.

math.CA↗
arXiv2009

The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Pólya-Schur on univariate polynomials with such properties.

math.CV↗
arXiv2008

Lee-Yang Problems and The Geometry of Multivariate Polynomials

We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory. This is an announcement with some of the main results in arXiv:0809.0401 and arXiv:0809.3087.

math.CV↗
arXiv2008

The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.

math.CV↗
arXiv2008

A converse to the Grace--Walsh--Szegő theorem

We prove that the symmetrizer of a permutation group preserves stability of a polynomial if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szegő Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the \emph{Grace-like polynomials} introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

math.CV↗
arXiv2008

Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

For $n \times n$ matrices $A$ and $B$ define $$η(A,B)=\sum_{S}\det(A[S])\det(B[S']),$$ where the summation is over all subsets of $\{1,..., n\}$, $S'$ is the complement of $S$, and $A[S]$ is the principal submatrix of $A$ with rows and columns indexed by $S$. We prove that if $A\geq 0$ and $B$ is Hermitian then (1) the polynomial $η(zA,-B)$ has all real roots (2) the latter polynomial has as many positive, negative and zero roots (counting multiplicities) as suggested by the inertia of $B$ if $A>0$ and (3) for $1\le i\le n$ the roots of $η(zA[\{i\}'],-B[\{i\}'])$ interlace those of $η(zA,-B)$. Assertions (1)-(3) solve three important conjectures proposed by C. R. Johnson 20 years ago. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials and as an application we derive similar properties for symmetrized Fischer products of positive definite matrices. We also obtain Laguerre type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices that considerably generalize a certain subset of the Hadamard-Fischer-Koteljanskii inequalities for principal minors of positive definite matrices. Finally, we propose Lax type problems for real stable polynomials and mixed determinants.

math.SP↗
arXiv2008

Negative dependence and the geometry of polynomials

We introduce the class of {\em strongly Rayleigh} probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons' recent results on determinantal measures and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.

math.PR↗
arXiv2008

Polya-Schur master theorems for circular domains and their boundaries

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region $Ω\subseteq \mathbb{C}$ for arbitrary closed circular domains $Ω$ (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for $Ω=\mathbb{R}$ our results settle open questions that go back to Laguerre and Pólya-Schur.

math.CV↗
arXiv2007

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * We characterize multivariate multi-affine polynomial with real coefficients that have the half-plane property (with respect to the upper half-plane) in terms of inequalities. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. * We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.

math.CO↗
arXiv2007

Actions on permutations and unimodality of descent polynomials

We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which strengthens recent unimodality results of Bóna. We prove that the generalized permutation patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingr\'ımsson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the $(P,ω)$-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. When restricted to the set of stack-sortable permutations we recover a result of Kreweras.

math.CO↗
arXiv2004

Counterexamples to the Neggers-Stanley conjecture

The Neggers-Stanley conjecture (also known as the Poset conjecture) asserts that the polynomial counting the linear extensions of a partially ordered set on $\{1,2,...,p\}$ by their number of descents has real zeros only. We provide counterexamples to this conjecture.

math.CO↗
arXiv2003

Finite automata and pattern avoidance in words

We say that a word $w$ on a totally ordered alphabet avoids the word $v$ if there are no subsequences in $w$ order-equivalent to $v$. In this paper we suggest a new approach to the enumeration of words on at most $k$ letters avoiding a given pattern. By studying an automaton which for fixed $k$ generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula \cite{Reg1998} for exact asymptotics for the number of words on $k$ letters of length $n$ that avoids the pattern $12...(\ell+1)$. Moreover, we give the first combinatorial proof of the exact formula \cite{Burstein} for the number of words on $k$ letters of length $n$ avoiding a three letter permutation pattern.

math.CO↗
arXiv2003

On operators on polynomials preserving real-rootedness and the Neggers-Stanley Conjecture

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.

math.CO↗
arXiv2002

q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$

The Narayana numbers are $N(n,k) = {1 \over n}{n \choose k}{n \choose {k+1}}$. There are several natural statistics on Dyck paths with a distribution given by N(n,k). We show the equidistribution of Narayana statistics by computing the flag h-vector of $J({\bf 2} \times {\bf n})$ in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana statistics in question have a distribution given by Fürlinger and Hofbauers q-Narayana numbers. We also interpret the h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions.

math.CO↗