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Re: Eliminate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116178] Re: Eliminate
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 4 Feb 2011 01:39:19 -0500 (EST)

Francisco Javier Garc=EDa Capit=E1n wrote:
> Hello, as I reposted on the forum, what I really wanted is to eliminate
> v and w

(I knew that.)


> (in trying to do this, I put v=1-t, w=t, this is why I did the
> mistake).


You can do this as follows. Subtract one side from the other to get
polynomials. Then extract a Groebner basis with an elimination ordering.

polys == {(v + w)*(b^2*v^4 - a^2*v^3*w + 3*b^2*v^3*w +
     c^2*v^3*w - 3*a^2*v^2*w^2 +
     3*b^2*v^2*w^2 + 3*c^2*v^2*w^2 - a^2*v*w^3 + b^2*v*w^3 +
     3*c^2*v*w^3 + c^2*w^4) - x,
   -(w*(-(b^2*v^4) - 2*b^2*v^3*w - a^2*v*w^3 + b^2*v*w^3 +
      c^2*v*w^3 + c^2*w^4)) - y,
  -(v*(b^2*v^4 - a^2*v^3*w + b^2*v^3*w + c^2*v^3*w -
     2*c^2*v*w^3 - c^2*w^4)) - z};

Either of these works. The first is faster, in your example. Isn't
always that way, though.

Timing[elim2 == First[GroebnerBasis[polys, {x,y,z}, {v,w},
   MonomialOrder->EliminationOrder,
   CoefficientDomain->RationalFunctions]];]

Timing[elim == First[GroebnerBasis[polys /. {a->4,b->-7,c->11},
   {x,y,z}, {v,w}, MonomialOrder->EliminationOrder]];]


> By the way, I contacted with Bernard Gibert and he told me that the
> equation that I am looking for is the quintic Q077
>
> http://mail.google.com/mail/?shva==1#inbox/12deb2a54c93072f
>
> and I have checked that the equation is that can be downloaded there:
>
> http://bernard.gibert.pagesperso-orange.fr/curves/Resources/Q077.rtf
>
> I am still interested in how Mathematica can arrive from parametric
> equations in terms of v,w  to a implicit equation in x,y,z (a,b,c here
> are constants)

As above. If you give explicit integer or rational values for those
constants then it becomes hugely faster, I might add.

Daniel Lichtblau
Wolfram Research



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