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.

💡 Research Summary

The paper establishes a powerful and unifying framework that connects totally non‑negative (TN) lower unitriangular matrices with families of real‑rooted polynomials, and then exploits this connection to solve a variety of problems in poset theory, algebraic combinatorics, and the theory of hyperplane arrangements. The authors begin by showing that any lower unitriangular matrix whose diagonal entries are all one is totally non‑negative if and only if it can be realized as a weighted path matrix R(Γ,λ) of a certain directed graph Γ equipped with non‑negative edge weights λ. This representation, together with Whitney’s reduction theorem, yields a unique array λ satisfying a natural monotonicity condition.

Next they introduce the notion of a “resolvable” matrix. A matrix R is resolvable when there exists a non‑negative weight array λ and a family of monic polynomials Rₙ,ₖ(t) satisfying a recursive relation that mimics the row‑generation process of R. The key equivalence (Theorem 2.6) states that for a lower unitriangular matrix the following three statements are equivalent: (i) R is resolvable, (ii) each row‑generating polynomial Rₙ(t) can be factored as (t+α₁)(t+α₂)…(t+αₙ) with non‑negative linear operators αᵢ, and (iii) R is totally non‑negative. When the αᵢ are non‑negative, every factor (t+αᵢ) has its zero in the interval