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