Subject: Rational function parametric representations
From: andre maute
Date: Wed, 23 Sep 2009 11:35:28 +0200
On Tuesday 22 September 2009, Barton Willis wrote:
> 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.
>
> Maple has a function that tries to find these parametric representations,
> but the user documentation doesn't give a reference to the algorithm.
There is a simple black box algorithm using Groebner Bases
on page 130 in the book
Cox, Little, O'Shea
Ideals, Varieties, and Algorithms,
An introduction to Computational Algebraic Geometry
and Commutative Algebra, 2nd ed.
Andre