Basics of Matroid Theory

For polynomials with nonnegative coefficients, we are introducing the concepts of strong and complete log-concavity. The main task for this week is proving Theorem 2.30, which offers us a new way to characterize Lorentzian polynomials.

Theorem 2.30 tells us that for a homogeneous polynomial \(f\) with nonnegative coefficients, the following are equivalent:

• \(f\) is comletely log-concave.
• \(f\) is strongly log-concave.
• \(f\) is Lorentzian.

We will prove the theorem using Proposition 2.33 as a lemma. By definition, we already know that (i) implies (ii). Therefore, it suffices to show that (ii) implies (iii), and (iii) implies (i). The key to proving the former is using the lemma to establish that \(f\) is \(2(1-1/d)\)-Rayleigh. For the latter, we observe that if \(f\) is Lorentzian (respectively \(logf\) is concave), \((\Sigma_{i=1}^{n}a_i\partial_i)f\) is Lorentzian (respectively \(\log(\Sigma_{i=1}^{n}a_i\partial_i)f\) is concave).