How to solve functional equation?

Question

B(2z^4 + 4z^6 + 9z^8 + 20z^{10} + 44z^{12} + 96z^{14}) =  (B(z))^4

I try to solve it using Maxima CAS :

(%i2) e: B(2*z^4 + 4*z^6 + 9*z^8 + 20*z^10 + 44*z^12 + 96*z^14) =  (B(z))^4;
                 14       12       10      8      6      4     4
(%o2)      B(96 z   + 44 z   + 20 z   + 9 z  + 4 z  + 2 z ) = B (z)
(%i3)  funcsolve (e,B(z));

expt: undefined: 0 to a negative exponent.
#0: rform(%r=[0,0])
#1: funcsol(%a=B(96*z^14+44*z^12+20*z^10+9*z^8+4*z^6+2*z^4) = B(z)^4,%f=B(z),l%=[])
#2: funcsolve(%a=B(96*z^14+44*z^12+20*z^10+9*z^8+4*z^6+2*z^4) = B(z)^4,%f=B(z))
#3: funcsolve(_l=[B(96*z^14+44*z^12+20*z^10+9*z^8+4*z^6+2*z^4) = B(z)^4,B(z)])
 -- an error. To debug this try: debugmode(true);

Here simpler example :

 define(f(z),z^2-1)
(%o3) f(z):=z^2-1
(%i4) f2:factor(f(f(z)))
(%o4) z^2*(z^2-2)
(%i5) e:B(f2) = B(z)^2
(%o5) B(z^2*(z^2-2)) = B(z)^2
(%i6) s:funcsolve(e,B(z))
expt: undefined: 0 to a negative exponent.
#0: rform(%r=[0,0])
#1: funcsol(%a=B(z^2*(z^2-2)) = B(z)^2,%f=B(z),l%=[])
#2: funcsolve(%a=B(z^2*(z^2-2)) = B(z)^2,%f=B(z))
#3: funcsolve(_l=[B(z^2*(z^2-2)) = B(z)^2,B(z)])
 -- an error. To debug this try: debugmode(true);

How should I do it? Is it another software / method for it ?

This is probably a pretty difficult problem. My advice is to ask about it on math.stackexchange.com. For Maxima, at least, I think you would have to work on it in steps; I doubt if Maxima can solve it from beginning to end. I don't know what's possible with other systems. — Robert Dodier, Jul 07 '22 at 18:25
https://marksmath.org/classes/Spring2019ComplexDynamics/text/section-super_attractive_conjugation.html — Adam, Jul 07 '22 at 19:41
@RobertDodier does it mean that I use funcsolve properly but it fails ? — Adam, Jul 07 '22 at 19:45
I'm not familiar with funcsolve, so my comments are based on general considerations about the problem, and general considerations about Maxima. I'm sorry I can't be more helpful. — Robert Dodier, Jul 07 '22 at 20:06

How to solve functional equation?

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