The Strongest Mason's Conjecture

This week, we mainly discussed the Strong Mason's Conjecture.

We aimed to prove the Strongest Mason's Conjecture (Theorem 4.14), which asserts that for any matroid \(M\), the sequence of the number of independent sets of size \(k\), denoted as \(I_k(M)\), is ultra log-concave. Our main goal was to understand how to translate this purely combinatorial counting problem into a problem related to Lorentzian polynomials.

We began with defining the homogeneous generating polynomial for independent sets \(f_M\) and then collapsed this into a bivariate polynomial where the coefficients are exactly \(I_k(M)\). If \(f_M\) is a Lorentzian polynomial, its coefficients automatically form an ultra log-concave sequence. Thus, proving the conjecture reduces entirely to proving that \(f_M\) is Lorentzian.

To prove \(f_M\) is Lorentzian, we introduced Tutte polynomial \(Z_{q, M}\) and the relation between them. Because the set of Lorentzian polynomials is closed, we only need to prove that \(Z_{q, M}\) is Lorentzian for \(0 < q \le 1\), which is Theorem 4.10.

Finally, we utilized the remarkable property that taking a partial derivative \(\partial_i\) corresponds combinatorially to the contraction of the matroid \(M/i\) and showed the Lorentzian nature of \(Z_{q, M}\) by induction.