I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly antiquated, definition of real algebraic singularities:
A point $P$ on a curve $A$ is said to be nonsingular if there exist open sets $U,V\in\mathbb{R}^d$ containing $P$ and an analytic isomorphism (bijective function analytic both ways) $f:U\rightarrow V$ such that $$f(A\cap U)=\{(x_1,x_2,...,x_r,0,0,...,0);x_i\in\mathbb{R}\}\cap V,$$ where $r$ is a nonnegative integer.
So to gain some intuition for this definition, I took a look at the two curves in $\mathbb{R}^2$: \begin{align} &y^3-x^2=0\\ &x^2-(y+1)^3+(y+1)^2=0. \end{align}
The first of these curves has a singularity at $(0,0)$, while the second one doesn't. The strategy the book uses in its examples is basically to apply the identity function to $x$, and then project $y$ down onto the $x-$axis, in the following way: \begin{align} &f(x,y)=0\\ &u=x\\ &v=f(x,y) \end{align}
Of course this depends on being able to invert the $v=f(x,y)$ function and solve for $y$. And in fact this is what fails for $y^3-x^2=0$, since solving $v=y^3-x^2$ for $y$ gives us $y=\sqrt[3]{u^2+v}$, which is not analytic at $(0,0)$.
So turning to the second equation, which is nonsingular at $(0,0)$, our second analytic function is $v=x^2-(y+1)^3+(y+1)^2$. Solving this equation for $y$ involves finding the roots of a cubic polynomial, which is not feasible for me. Thus if this second equation is to be nonsingular, then should I expect that one of these three roots will be a real-valued analytic function inside some open ball containing the origin, while the other two solutions will be complex-valued always with non-zero imaginary part?
If not, then how would I go about finding this analytic isomorphism?
I think the point is that you don't need to find an explicit formula for the roots. You don't need to worry about the roots that are not near $(0,0)$. A (complex) zero in $y$ of an analytic function $f(y,z)$ is an analytic function of $z$ in a neighbourhood of any point where the zero has multiplicity $1$.