From: saxena@ramanujan.cs.albany.edu (Tushar Saxena)
Newsgroups: sci.math.symbolic
Subject: Re: Q: Mma (or other) Eliminate
Date: 24 Jan 1997 16:33:32 GMT

The following variable elimination problem is easily solved by using
multivariate resultants.  One of the method, which we have found to
be quite efficient, is based on the Dixon formulation.  See the following
papers (available from http://www.cs.albany.edu/~saxena) to read about it:

Kapur D., Saxena T. and Yang L., Algebraic and Geometric Reasoning
using Dixon Resultants, {\em Proc. ACM International Symposium on
Symbolic and Algebraic Computation}, Oxford, England, July 1994.\\
 
Kapur D and Saxena T., Comparison of Various Multivariate Resultant
Formulations, {\em Proc. ACM International Symposium on Symbolic and
Algebraic Computation}, Montreal, July 1995.\\
 
Kapur D and Saxena T., Sparsity Considerations in the Dixon Resultant
Formulation, {\em Proc. ACM Symposium on Theory of Computing},
Philadelphia, May 1996.\\

This method is quite easy to implement in your favourite computer
algebra system, and even a naive implementation should work surprisingly
well!

You said:

>Mathematica gives no answer after 24 hours in a Pentium or Sun:

I used my MAPLE implementation of this method to solve your problem
in less than 50 seconds.  And that was after substituting a symbolic
parameter for 3^(1/2).  The result was a degree 8 polynomial in $x,y$
and had 81 terms.  To save bandwidth, I am sending you the polynomial
by email.

Tushar Saxena
Research Consultant
General Electric, Corporate Research & Development

--------------------------------------------------

In article <32E372D7.158D@uvigo.es>,
Francisco Botana  <fbotana@uvigo.es> wrote:
>Trying to obtain in Mathematica 2.2.3 the envelope of a family of lines
>I've got some problems.
>
>If the lines are ellipses of expression
>
>with focus in the ellipse
>
>e1=-500 - 80*a + 4*a^2  + 9*b^2
>
>e22=-80 + 21*a^2 + 32*b + 31*b^2 - 32*3^(1/2)*x + 
>  10*3^(1/2)*b*x + 21*x^2 - 32*y - 62*b*y - 
>  10*3^(1/2)*x*y + 31*y^2 + 
>  a*(32*3^(1/2) - 10*3^(1/2)*b - 42*x + 10*3^(1/2)*y)
>
>Mathematica gives no answer after 24 hours in a Pentium or Sun:
>
>Eliminate[{e1==0,e22==0,
>			Dt[e1,a,Constants->{x,y,b}]*
>			Dt[e22,b,Constants->{x,y,a}]==
>			Dt[e22,a,Constants->{x,y,b}]*
>			Dt[e1,b,Constants->{x,y,a}]},
>		{a,b}]
>
>...
>
>My questions are:
>
>How can I speed up the answer (if Mma can obtain it)?
>Is there any other symbolic package capable of coping with this problem?
>How?
>
>Thanks in advance.
>-- 
>			   Francisco Botana
>   Depto. de Matematica Aplicada   EUIT Forestal-Universidad de Vigo
>   Campus de A Xunqueira, s/n      36005 PONTEVEDRA (Spain)
>   tel +34-86-801931   		   fax +34-86-801907