24 Van der Waerden’s permanent conjecture
Let \(M = (m_{ij})\) be a doubly stochastic \(n \times n\) matrix. Then
\[ \operatorname {per}M \ge \frac{n!}{n^n} \]
and equality holds if and only if \(m_{ij} = \frac{1}{n}\)
Proof
TODO
If \(p(x)\in \mathbb {R}_+[x_1, \dots , x_n]\) is a \(H\)-stable and homogeneous of degree \(n\), then either \(p'\cong 0\), or \(p'\) is \(H\)-stable and homogeneous of degree \(n - 1\). In either case
\[ \operatorname {cap}(p') \ge \operatorname {cap}\cdot g(\deg _np). \]
Proof
TODO