Here comes a simple scenario
fz = 8 (z + 18 A z + (8 - 48 A) z^3 + 16 z^5); sol = Solve[fz == 0, z]
As we can see there are five roots. The root (0,0) is always present while the nature of the other four roots strongly depends on the numerical value of A. I found that when $A < -1/18$ we have two real and two purely imaginary roots. For $A > -1/18$ there are three cases:
(i) When $A \in (-1/18, A_1)$ we have 4 purely imaginary roots,
(ii) When $A \in (A_1, A_2)$ we have 4 complex roots,
(iii) When $A > A_2$ we have 4 real roots.
My question: How can I determine the exact values of $A_1$ and $A_2$, whiche delimit the three intervals?
Many thanks in advance!