From Lemma 2.8 and 2.9 to Theorem 2.13

Lemma 2.8 and 2.9 to Theorem 2.13

• Lemma 2.8 ensures that the operator \(1 + θ w_i ∂_j\) preserves \(M\)-convex support.
• Lemma 2.9 provides stability and interlacing properties that control how derivatives behave under this operator.

Using these two lemmas, Proposition 2.7 shows that this operator preserves the Lorentzian property.

Theorem 2.10

The crucial step is Theorem 2.10, which upgrades this local operator result to invariance under general nonnegative linear changes of variables. This gives strong structural stability of the Lorentzian class. In particular, Corollary 2.11 shows that nonnegative directional derivatives preserve Lorentzian polynomials.

Final Step

With this structure in place, Lemma 2.12 constructs a Nuij-type homotopy that pushes a Lorentzian polynomial with positive coefficients into the strictly Lorentzian region. Finally, Theorem 2.13 combines the nonnegative linear change of variables and the homotopy to show that every Lorentzian polynomial can be approximated by strictly Lorentzian ones, proving the closure result.

Remark