I am looking for help to understand when Puiseux series must be used. I have the follow function:
$$x^2 z^4 (x^2-(z-k)^2)=(z-k)^4(z^2-1)$$
When I use mathematica AsymptoticSolve, I get Taylor series expansion:
$$z(x) \approx k + \sqrt{\frac{k^2 \left(k^2+\sqrt{k^4+4k^2-4}\right)}{2(1-k^2)}} x$$
which has a singularity at $k=1$.
When I set $k=1$ to the above expression and using AsymptoticSolve again, I get Puiseux series expansion:
$$z(x) \approx 1-\left(\frac{x}{2}\right)^{2/3}+\frac{11}{6}\left(\frac{x^2}{2}\right)^{2/3}$$
Is there anyone can help me to understand why $k=1$ is special in the above equation such that it turns out to be singularity on Taylor series expansion and require Puiseux series expansion.
Is it possible to force Puiseux series expansion for unspecified $k$ and remove the singularity?
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Edit: I study the equation further. It seems that when $k=1$, the expansion becomes singular perturbation since I'll lose 4 roots using regular perturbation. This however is slightly different from normal singular perturbation where $x$, not $k$, is assumed to be small. Here $k$ is the coefficient instead and is not small. In such situation, any chance I can find an expansion of $z$ that applies for all $k$? If I treat $z$ a function of $k$ instead, any kind of expansion that would uniformly converge for $k \in [0,2]$