Using Maple to find Groebner Bases and Implicit Equations

First you have to fetch the Groebner Package in Maple

>    with(Groebner);

[MulMatrix, SetBasis, fglm, gbasis, gsolve, hilbertdim, hilbertpoly, hilbertseries, inter_reduce, is_finite, is_solvable, leadcoeff, leadmon, leadterm, normalf, pretend_gbasis, reduce, spoly, termorder...
[MulMatrix, SetBasis, fglm, gbasis, gsolve, hilbertdim, hilbertpoly, hilbertseries, inter_reduce, is_finite, is_solvable, leadcoeff, leadmon, leadterm, normalf, pretend_gbasis, reduce, spoly, termorder...

Now you define your polynomials

Say we are given x(t)=t^2-2t and y(t)=(t^3+1)/(t-2)

>    f1:=t^2-2*t-x; f2:= t^3+1-y*(t-2); f3:= s*(t-2)-1;

f1 := t^2-2*t-x

f2 := t^3+1-y*(t-2)

f3 := s*(t-2)-1

Now we can find our basis using the lex order s>t>x>y

>    gbasis([f1,f2,f3],plex(s,t,x,y));

[-18*y-16*y*x+x^3-2*y*x^2-6*x-9+y^2*x, x^2-7*t-y*x-4*x-4+4*y*t-8*y, 4*t*x+x^2+9*t-y*x+4*x, t^2-2*t-x, 4*t+x+9*s-y+4]

Look at this ideal and take out the polynomial in only x and y.

We find that the implicit form of the curve is defined by  f = -18*y-16*y*x+x^3-2*y*x^2-6*x-9+y^2*x