I'm interested in the following question:
Could a curve where both ends approach each other (see below for illustration), (always) be described using the zero set of a polynomial $P(x,y)$ of two real variables?
My intuition tells me this might be impossible, but on the other hand, this particular image is the zero set of $$y^2-x \mathrm{e}^{-x}. $$ Thus, perhaps some modification of the exponential here could give such an example.
In general, I'd like to know if there is any simple characterization of all plane curves which are the zero sets of some polynomial $P(x,y)$. Thank you.
How about the zero set of $y^2(x^2+1) - x$?
This can be rewritten as $y^2 = \frac{x}{x^2+1}$. (It's just your example, but with $e^x$ replaced by $x^2+1$, a polynomial which also shows the relevant behavior of taking values $\geq 1$ for nonnegative $x$ and growing faster than $x$.) As $x$ gets large, the fraction on the right gets very small, so $y$ gets very small and either positive or negative, i.e. "the ends approach each other."
I very much doubt that you're going to find a characterization of plane curves defined by polynomials by properties like this that you can "see" by looking at a picture.