Suppose I have an implicit equation:
\begin{equation} f(x_0,x_1,\ldots,x_n)=0 \end{equation}
Which might be 'paramaterizable'; i.e. put it into the form: \begin{align} x_0 &= g_0(t_0,t_1,\ldots,t_m) \\ x_1 &= g_1(t_0,t_1,\ldots,t_m) \\ \vdots \\ x_n &= g_n(t_0,t_1,\ldots,t_m) \end{align}
where each of the $g$ functions may be in terms of any of the parameters $t_k$ up to $t_m$, where $m \le n$.
I have two queries, which I suspect are closely related:
- Is there a general test which can be performed to determine if parameterization is even possible? I already know from this answer that not all curves can be parameterized, but I don't quite understand how to test this.
- Judging by the fact that this question does not yet have an answer, it suggests that there is no algorithm for determining the parameterization of a function $f$. I found three examples of parameterization of specific functions on this site (a trochoid, power function, and a line). The first two answers use heuristics very specific to the problems in question, and the last one uses a general rule for two variables, when one can be written explicitly (when $y=f(x)$, $x=t$ and $y=f(t)$, quite trivial). So if there is no algorithm, is there a widely accepted heuristic approach to parameterization (i.e. something akin to the LIATE rule for integration by parts)? The more specific, the better.