Let $f:\mathbb{R}^3\rightarrow\mathbb{R}$ be a polynomial. What are some necessary and/or sufficient conditions, preferably on the polynomial coefficients, for a given isosurface of $f$ to be bounded?
Based on post What does it mean for a level curve to be closed or open? I conjecture that one example of a sufficient condition is $|f(x,y,z)|\rightarrow\infty$ as $|x|+|y|+|z|\rightarrow\infty$ but I am not sure if this is equivalent to some other condition related to the polynomial coefficients.
If the mentioned sufficient condition holds, does it hold for all isosurfaces of $f$?
On a related note, what are some necessary and/or sufficient conditions for an isosurface of $f$ to consist of a single connected component?