Consider $$\textbf{F}(x,y,z) = ((x+z)^2, y-z) = 0$$ Taking the Jacobian at the point $(0,0,0)$ gives a matrix of rank $1$, which from my understanding means that the implicit function theorem cannot guarantee an implicit function $g(z)=(x,y)$ in the neighborhood $(0,0,0)$.

However, the book I am reading suggests that an implicit function $\textbf{does}$ exist, in the form of $g(z)=(-z,z)$, and concludes that $\textbf{F}(g(z),z)=(0,0)$ in a neighborhood of $(0,0,0)$.

I really don't understand the intuitive idea and result here at all (and maybe I'm completely misunderstanding the implicit function theorem). I know that the implicit function theorem shows when we can write $n-k$ "passive variables" as a function of the one "free variable" (in this case $z$). I have some questions:

  1. Is the implicit function theorem a sufficient condition but not a necessary condition for finding an implicit function $g$ of the $k$ free variables at a point? How do we know $g(t)=(-z,z)$ satisfies the conditions stated in a "neighborhood of $(0,0,0)$"?
  2. If so, how can I reconcile these two results with each other? What is missing from the implicit function theorem?

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