Solution
= Solution
Let
$$
F(x,y,z)=x^2+y^2+z^3+az+b.
$$
By the <regular level set theorem>, $F^{-1}(0)$ is a smooth surface wherever $\nabla F\ne0$. A critical point on the level set would satisfy
$$
x=y=0,
\qquad a=-3z^2,
\qquad b=2z^3.
$$
These equations imply $4a^3+27b^2=0$. Therefore
$$
\boxed{4a^3+27b^2\ne0\Longrightarrow S_{a,b}\text{ is a smooth surface}.}
$$