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How would I design a non-trivial algebraic function of degree 4 containing a branch at the origin with the (finite) power expansion:

##w(z)=1+0.5 z-1/4 z^{1/2}+3/4 z^{1/4}##?

having the form

## f(z,w)=a_0+a_1 w+a_2 w^2+a_3 w^3+a_4 w^4=0##

with the ##a_i ## ( preferably not fractional) polynomials? And if that is possible, can I design any function ##w(z)## with a finite power expansion with ##f## of any degree?

I don't know.

##w(z)=1+0.5 z-1/4 z^{1/2}+3/4 z^{1/4}##?

having the form

## f(z,w)=a_0+a_1 w+a_2 w^2+a_3 w^3+a_4 w^4=0##

with the ##a_i ## ( preferably not fractional) polynomials? And if that is possible, can I design any function ##w(z)## with a finite power expansion with ##f## of any degree?

I don't know.

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