Theorem 4.28 (Marcus, Spielman, Srivastava (real rooted)). For every $ρ_{1}, \dots, ρ_{m} \in [- 1, 1]$ the following polynomial is real rooted

χ_{ρ_{1}, \dots, ρ_{m}} (x) = \sum_{S \in {\pm 1}^{m}} \det (x I - A_{s}) \prod_{J : s_{J} = 1} (\frac{1 + ρ_{J}}{2}) \prod_{J : S_{J} = - 1} (\frac{1 - ρ_{J}}{2}) .