TITLE: Applications of Gröbner bases
QUESTION [19 upvotes]: I would like to present an application of Gröbner bases.  The audience is a class of first year graduate students who are taking first year algebra.  
Does anyone have suggestions on a specific application that the audience would appreciate?

REPLY [6 votes]: find intersection points of a couple of conics (pick the right coefficients to make it not so tedious to do all the manipulation)
describing the motion of a constrained single hinged robot arm or planetary epicycles (make a cardioid from two equations)
colorability of a graph (see A Crash Course... ) (when presented with the construction, very easy to see that the algorithm produces a solution)

REPLY [5 votes]: I learnt of a cool application here in Math.SE where I had asked a question to parametrize $$x=2t-4t^3$$ $$y=t^2-3t^4$$
There was no straightforward way to eliminate $t$, however a user pointed out

using a Gröbner basis routine such as that in Mathematica easily gives the implicit Cartesian equation
$$27x^4-4x^2(36y+1)+16y(4y+1)^2=0$$
In Mathematica: GroebnerBasis[{x == 2t - 4t^3, y == t^2 - 3t^4}, {x, y}, t]

I doubt this would be fascinating to graduates though.