[Paper Review] On linear transformations preserving the Pólya Frequency property

Petter Brändén|arXiv (Cornell University)|Mar 22, 2004

Advanced Combinatorial MathematicsMathematics21 citations

TL;DR

This paper establishes sufficient conditions for linear operators to preserve the Pólya frequency (PF) property and real-rootedness of polynomials, using differential operators and bilinear forms. It resolves open conjectures in combinatorics, including the real-rootedness of $q$-Eulerian polynomials and $h$-vectors of Fomin-Zelevinsky simplicial complexes, and proves that certain generating functions of permutation statistics are PF sequences.

ABSTRACT

We prove that certain linear operators preserve the Pólya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and Reiner-Welker.

Motivation & Objective

• To characterize linear operators that preserve the Pólya frequency (PF) property and real-rootedness of polynomials.
• To resolve open problems in combinatorics concerning the real-rootedness of generating functions for permutation statistics.
• To generalize classical theorems of Hermite, Pólya, Schur, and Wagner by introducing sufficient conditions on generating functions of differential operators.
• To establish that certain polynomial bases transform PF sequences into real-rooted polynomials under the $τ$-operator.
• To prove that $h$-vectors of simplicial complexes associated with finite Weyl groups are PF, settling a conjecture by Reiner and Welker.

Proposed method

• Uses differential operators of the form $\phi_F = \sum_{k=0}^n Q_k(x) \frac{d^k}{dx^k}$, where $F(x,z) = \sum_{k=0}^n Q_k(x) z^k$, to analyze PF-preserving transformations.
• Applies the theory of total positivity and interlacing of real-rooted polynomials to derive conditions under which such operators preserve the PF property.
• Employs the $\mathcal{E}$-operator defined by $\mathcal{E}(\binom{x}{i}) = x^i$ to transform basis expansions into standard monomial forms.
• Utilizes the Obreschkoff theorem and properties of multiplier sequences to analyze interlacing and real-rootedness of linear combinations of polynomials.
• Applies Theorem 4.6 to show that if a polynomial has nonnegative coefficients in the basis $\{x^i(x+1)^{d-i}\}$, then its $\mathcal{E}$-image is real-rooted.
• Uses the Hadamard product and known results on real-rootedness under coefficient-wise multiplication to analyze $h$-polynomials of simplicial complexes.

Experimental results

Research questions

• RQ1 Under what conditions does a linear differential operator $\phi_F$ preserve the Pólya frequency property of a sequence?
• RQ2 Are the $q$-Eulerian polynomials $A_n(x;q)$ real-rooted for all integers $q$?
• RQ3 Do the $h$-vectors of Fomin-Zelevinsky simplicial complexes associated with finite Weyl groups form PF sequences?
• RQ4 Is the generating function for $t$-stack sortable permutations with $k$ descents a PF sequence for $t=2$ and $t=n-2$?
• RQ5 Does the polynomial $P(B_n, S; x) = \sum_{\sigma \in B_n, N(\sigma) \in S} x^{d_B(\sigma)}$ have only real and simple zeros for any subset $S \subseteq [0,n]$?

Key findings

• The $q$-Eulerian polynomials $A_n(x;q)$ are real-rooted for all integers $q$, confirming a conjecture by Brenti.
• The $h$-vector of the Fomin-Zelevinsky simplicial complex $\Delta_{FZ}(D_n)$ is a PF sequence, resolving an open problem by Reiner and Welker.
• The generating function for the number of $t$-stack sortable permutations in $\mathcal{S}_n$ with $k$ descents is a PF sequence for $t=2$ and $t=n-2$, confirming two new cases of Bóna's conjecture.
• For any subset $S \subseteq [0,n]$, the polynomial $P(B_n, S; x)$ has only real and simple zeros, extending a result of Brenti.
• The polynomial $F_n(\alpha,\beta) = \alpha h(\Delta_{FZ}(B_n),x) + \beta n x h(\Delta_{FZ}(A_{n-2}),x)$ is real-rooted and simple for all $\alpha \geq 0$, $2\alpha + \beta > 0$, with strict interlacing or alternation properties.
• The $\mathcal{E}$-image of any polynomial with nonnegative coefficients in the basis $\{x^i(x+1)^{d-i}\}_{i=0}^d$ is real-rooted with only real, non-positive, and simple zeros.