Dowling's polynomial conjecture for independent sets of matroids
Shiqi Cao, Keyi Chen, Yitian Li, Yuxin Wu
TL;DR
This work resolves Dowling's polynomial conjecture by embedding the problem in the theory of Lorentzian polynomials. By considering the Lorentzian structure of $G_M(oldsymbol{x})G_M(oldsymbol{y})$ and applying careful linear operators, the authors derive a key Hessian-based inequality that yields $f_k^2(M) \,\ge\,\left(1+\frac{1}{k}\right) f_{k-1}(M) f_{k+1}(M)$ for $0<k<r_M$, proving Dowling's polynomial conjecture and Zhao's strengthened form; they further establish a broad generalization to $f_l^p(M)$ inequalities. The results are recast in terms of counts of independent partitions $\,\pi_{i_1,...,i_p}(N)$ for minors $N$ and shown to be equivalent to coefficient inequalities in associated polynomials, with Lorentzian-structure preserved under the necessary operators. The last section extends the framework to a multi-parameter setting, linking product inequalities of $f$-values to partition-count monotonicity on suitably constructed minors $M[oldsymbol{X},oldsymbol{q}]$, thereby unifying and broadening the Mason-type inequalities for matroid independent sets.
Abstract
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.