Proposition 4.32 (Stability of real stable). Assuming that:

Then the following are also:

$p (z_{σ (1)}, \dots, z_{σ (n)})$ , $σ \in S_{n}$ .
$p (a z_{1}, z_{2}, \dots, z_{n})$ , $a \in ℝ_{\geq 0}$ .
$p (z_{2}, z_{2}, z_{3}, \dots, z_{n})$ .
$p (c, z_{2}, \dots, z_{n})$ , $c \in H \cup ℝ$ .
$z_{1}^{d_{1}} p (- \frac{1}{z_{1}}, z_{2}, \dots, z_{n})$ .
$\partial_{z_{1}} p (z_{1}, \dots, z_{n})$ .
$MAP (p (z_{1}, \dots, z_{n}))$ . $MAP (z_{1} z_{2} + z_{2} z_{3}^{2} + 2 z_{1} z_{4} + z_{2} z_{3}^{2}) = z_{1} z_{2} + 2 z_{1} z_{4}$ .
If $p, q$ are real stable, then so is $p q$ .