Artin approximation vs implicit function theorem in the class of analytic functions I asked this on math stackexchange but I had no luck, so I am posting my question also here.
I am not an algebraist so my question might be stupid. I am doing mainly complex analysis and recently I was informed about the existence of Artin's theorem and it sounded like it could be of interest to me. I have found a survey on the subject and I started reading it. Here's the link.
So to the actual theorem (section 1.1)

Let $\mathbb{k}$ be a field of characteristic 0 and let $f(x,y)$ be a vector of convergent power series in two variables $x$ and $y$. Assume given a formal power series $\hat{y}(x)$ vanishing at 0,
  $$f(x,\hat{y}(x))=0.$$
  Then for any $c\in\mathbb{N}$, there exists a convergent power series solution $\tilde{y}(x)$
  $$f(x,\tilde{y}(x))=0$$
  which coincides with $\hat{y}(x)$ up to degree $c$,
  $$\hat{y}(x)\equiv \tilde{y}(x) \mbox{ mod }x^c. $$

I really care only for the case where $k=\mathbb{C}$. Using the implicit function theorem for some analytic $f$ we get the existence of an analytic solution as long as the Jacobian has full rank at 0. If on the other hand the Jacobian does not have full rank then we generically get some king of branching and this means that there is no formal solution in powers of $x$ that solves the equation. So in that sense I don't see how Artin's theorem is stronger than the implicit function theorem in the analytic setting.
Is this true or do I miss something? By the way I don't know what happens when we consider other fields and I don't imply that the theorem is trivial or useless. 
EDIT: As was pointed out correctly by wrigley the implicit function theorem can fail at points where there are multiple solutions. For example when $f(x,y)=x^2-y^2$. In this case the workaround is to consider one solution at a time, i.e. factorize $f$ and look at one factor each time.
 A: In the case when y is a single variable the set of formal solutions is a finite set and Artin's Theorem implies that all these solutions are convergent. But this is a very particular case and not the most interesting application of this theorem. 
In the case when y is a set of several variables then the formal solutions are not always convergent. For example if you consider the equation $y_1^2-y_2^3=0$ as an equation with coefficients in $\mathbb{C}\{x\}$, the formal solutions are couples $(y_1(x),y_2(x))=(z(x)^3,z(x)^2)$ where $z(x)$ is a formal power series. But if $z(x)$ is not convergent then the corresponding solution is not convergent. On the other hand if you replace $z(x)$ by one of its truncations (you remove in $z(x)$ all the monomials of degree higher than a given number $c$), then you obtain a polynomial $z'(x)$ and the couple $(z'(x)^3,z'(x)^2)$ is a solution of the equation which is close to the given formal solution. And its components are convergent power series (here in fact they are even polynomials).
