Rational function parametric representations

Barton Willis a ?crit :
> To find a rational function parametric representation of a folium of 
> Descartes,
> there is (the cute trick) of using the substitution y = x * p:
>
>  (%i25) [x^3 + y^3 - 3*a*x*y, y = x * p];
>  (%o25) [y^3-3*a*x*y+x^3,y=p*x]
>
>  (%i26) algsys(%,[x,y]);
>  (%o26) [[x=(3*a*p)/(p^3+1),y=(3*a*p^2)/(p^3+1)],[x=0,y=0]]
>
> What is the generalization of this trick? Sometime ago, I saw an
> article on this, but I can't find it. Maybe I gave up to soon, but
> a web search on  "parametric representation" and "parametric 
> representation algorithm" didn't locate the paper.
>   
Maybe this book gives what you're looking for :
http://www.springer.com/math/algebra/book/978-3-540-73724-7

And this article :
http://www.risc.uni-linz.ac.at/publications/download/risc_271/Nr.12_winkler.pdf


An algebraic curve is rational (parametrizable by rational functions) 
only in some very special cases ; it has to be of genus 0. For instance, 
x^3+y^3=1 is not a rational curve.
In the case of a plane curve of degree n with a point of multiplicity 
n-1, the trick of moving a line through the multiple point gives a 
rational parametrization. For instance, take any point on the circle 
x^2+y^2=1, say (-1,0) then the line with slope t through this point cuts 
the circle in 2 points, of which one is known so the other one may be 
rationaly determined in terms of t. the result is the well known 
parametrization ((1-t^2)/(1+t^2) , 2t/(1+t^2)). For the same reason, any 
cubic with a double point is rational (folium of Descartes for instance) 
, and the trifolium (x^2 + y^2)^2 +a*x*(3*y^2-x^2) which is a quartic 
with a triple point is rational : any line through this point intersects 
the curve at a fourth point, rational in terms of the slope.

For other types of singularities, the parametrization seems to require 
more involved techniques. I can't help.
> Maple has a function that tries to find these parametric representations,
> but the user documentation doesn't give a reference to the algorithm.
>   

Does Maple solve for instance the case of the tacnode curve 
2*x^4-3*x^2*y+y^4-2*y^3+y^2 given in Winkler's article above ? This 
curve has two singularities.

Eric Reyssat