Solving for one of variables locally when the Implicit Function Theorem does not apply I'm having trouble deciding whether certain functions can be locally solved. I have some examples:


*

*Can $xye^{xz} - z\log y =0$ be locally solved in $(0,1,0)$ for x? y? z?


In this case, I used Implicit Function Theorem to answer affirmatively that $x$ can be solved locally in function of $y$ and $z$. As $d/dy$ and $d/dz$ are equal zero, I can't use the theorem in those cases. I tried proving that $x$ has a local maximum at $(0,1,0)$ and could prove the first derivatives of $x$ with respect to $y$ and $z$ are zero, but I couldn't go further. 


*

*If $f(x,y) = \frac{\sin(xy^2)}{xy}$ if $x\neq y$ and $f(x,y)=0$ otherwise, can $f(x,y) = 0$ be locally solved for $y=y(x)$ around $(0,0)$?


I couldn't apply implicit function theorem again and I have a clue that we can solve this locally for $y$, using $sin$ power series. 
Is there a general procedure for problems like that? Any help is appreciated. 

When implicit function theorem does not apply, the approach is usually ad hoc, based on the details of the function. Two things to try:

*

*Change the variables. Some substitutions, like $u=x^3$, can reduce the function to a form in which the implicit function theorem applies.

*Restrict $f$ to some line parallel to the coordinate of interest. If the zero set has multiple points, one can't solve for that coordinate.

First example
Restriction to lines works here:

*

*The line $x=0=z$ is in the zero set, which shows you can't solve for $y$

*The line $x=0$, $y=1$ is in the zero set, which shows you can't solve for $z$
Second example
The above tricks are not needed: the structure of particular function helps. Indeed, $f(x,y)$ turns into zero precisely when $\sin(xy^2)=0$. And since we are in a neighborhood of $(0,0)$, that means $xy^2=0$, which is equivalent to $x=0$ or $y=0$. Now the local structure of the set $f=0$ is clear.
