Is there a general method for trying to finding parametric equations for polynomial equations?
The classic example of $x^2 + y^2 = 1$ having multiple parameterizations:
\begin{array}{rcl} (x,y) &=& (\sin t, \cos t) \\ (x,y) &=& (\cos t, \sin t) \\ (x,y) &=& \left(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2}\right) \end{array}
which are easily seen to work by plugging in. Note that the first two involve elementary functions, but the last one only rational functions.
I'm curious about higher degree polynomials. Is there a general method to find a parameterized set of equations in rational functions? That is, is there an algorithm for extracting the parameter functions for $x$ and $y$ from the equation (or arbitrary sets of equations over $v\ge1$)? And if not, is there a characterization of which implicit sets can be parametrized?
Take the (random off the top of my head) example:
$$x y^3 + x^2 y + x y = 1$$
Is there a set $x(t), y(t)$ for this that can be found algorithmically? (I may well be naive here, not realizing something obvious, or maybe the entire field of algebraic geometry is geared towards possible solutions to this).
One can do the reverse process of creating an implicit set of equations via Gröbner basis calculation, trying to eliminate $t$. Or likewise in the forward direction if you can eliminate variables one-by-one with GB that would suffice to parameterize via one of the original variables. But in general if elimination isn't possible, is there still an algorithm to (even partially) parameterize?
