Is there a variant of the implicit function theorem covering a branch of a curve around a singular point?

I think you have to make a choice of whether you want the local behavior of the paramterized curve (a circle) or to study the self-intersecting behavior of your planar curve embedded in a copy of $\mathbb{C}$ (or choice of ambient space).

Wikipedia offers two different paramterizaions:

• polar coordintes $\displaystyle r = \frac{3a \, \sin \theta \cos \theta}{\sin^3 \theta + \cos^3 \theta} $ and expand around $\theta = 0, \frac{\pi}{2}$.
  
  
• rational functions $\displaystyle x = \frac{3at}{1 + t^3} $ and $\displaystyle y = \frac{3at^2}{1 + t^3} $ and expand around $t = $.

Singular points of planar curves have been studied for a long time, in classical language (using equations) and there books on the theory of algebraic curves. Yet... the mapping between the modern and classical language is lacking in my opionion.

• Hilton, Harold (1920). "Chapter II: Singular Points". Plane Algebraic Curves. Oxford.

Here he starts to outline the different kinds of singular points that may occur. One problem is that once you have singular points of any kind these are no longer varieties, they are schemes. Even though this very basic case has been studied since Isaac Newton.

Often modern elementary discusions of algebraic geometry avoid discussions of singular points altogether. I guess in order to make the discussion smooth they rule out all the interesting parts?

• Nigel Hitchin Algebraic Curves (Oxford, 2009)

• Frances Kirwan Complex Algebraic Curves

She does algeraic geometry over $\mathbb{C}$, talks about Puiseux series (to handle both branches) and advocates resolution of singularities.