Puiseux expansions at branch points

Find the first two terms in the expansion of $y$ in increasing powers of $x$ for the branches of the curve $x^{3}-3xy+y^{3} = 0$ passing through the origin.

Find the first two terms in the expansion of $y$ in increasing powers of $x$ for the branches of the curve $y^{4}-4y = x$ passing through the origin.

Find the singular points of the equation $y^{5}+ax^{4}-b^{2}xy^{2} = 0$.

Find the singular points of the equation $4(x-1)y^{5}+2xy^{3}-3x^{3}y+x^{4} = 0$.

Let $z = a$ be a branch point of order $n$ for the function $\mathcal{F}(z)$. Prove that the expression $\mathcal{F}\left(a+\zeta^{n}\right)$ represents $n$ functions having the point $\zeta = 0$ as an isolated singular point of single-valued character.

Prove that if the function $\mathcal{F}(z)$ has the point $z = a$ as a branch point of order $n$, then it can be represented in the form

where the function $f(w)$ has the point $w = 0$ as an isolated singular point of single-valued character.

Prove that a function $\mathcal{F}(z)$ having at the point $z = a$ an algebraic branch point of order $n$ can be represented in the form $\mathcal{F}(z) = f(\sqrt[n]{z-a})$, where the function $f(w)$ is regular at the point $w = 0$ or has a pole there.

Prove that a function $\mathcal{F}(z)$ having at the point $z = a$ an algebraic branch point of order $n$ can be represented in the form

where the functions $g_{k}(z)$ are regular at the point $z = a$ or have poles there.