Dowling's polynomial conjecture for independent sets of matroids

The celebrated Mason’s conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason’s conjecture was independently solved by Anari, Liu, Oveis Gharan and Vinzant, and by Brändén and Huh. The weak form of Mason’s conjecture was also generalized to a polynomial version by Dowling in 1980 by considering certain polynomial analogue of independent set numbers. In this paper we completely solve Dowling’s polynomial conjecture by using the theory of Lorentzian polynomials.

💡 Research Summary

The paper resolves Dowling’s polynomial conjecture, a 1980 proposal that strengthens Mason’s log‑concavity conjecture for matroids by replacing the numerical inequality on independent‑set counts with a polynomial inequality. For a matroid M on a ground set E, let
(f_k(M)=\sum_{I\in\mathcal I,|I|=k}\prod_{i\in I}x_i)
be the homogeneous degree‑k polynomial encoding all independent sets of size k. Dowling conjectured that for every 0 < k < r(M) one has
(f_{2k}(M)\ge f_{k-1}(M)f_{k+1}(M))
(the “weak” form). Zhao later suggested a slightly stronger version with a factor ((1+1/k)) on the right‑hand side. Both conjectures were known only for small k (≤7 for Dowling, ≤5 for Zhao).

The authors prove Zhao’s stronger inequality, which immediately implies Dowling’s original statement. Their approach rests on the modern theory of Lorentzian (equivalently, completely log‑concave or strongly log‑concave) polynomials introduced by Brändén–Huh and Anari‑Liu‑Oveis Gharan‑Vinzant. The key object is the polynomial
(G_M(x)=\sum_{I\in\mathcal I}x^{n-|I|}\prod_{i\in I}x_i)
where n=|E|. It was shown in earlier work that (G_M) is Lorentzian. Lorentzian polynomials enjoy two crucial properties: (i) their support is M‑convex, and (ii) for any multi‑index α with total degree d‑2, the Hessian of the partial derivative (\partial^{\alpha}G_M) has at most one positive eigenvalue. Moreover, the class is closed under non‑negative linear changes of variables, non‑negative directional derivatives, and multiplication.

The proof proceeds by considering the product (G_M(x)G_M(y)) in two independent sets of variables and applying the linear operator
(S_i = (\partial_{x_i}+\partial_{y_i})\big|_{x_i=y_i=0})
to each pair of variables, defining (S=S_1\cdots S_n). Lemma 3.1 shows that the coefficient (