Applications of Gröbner bases 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?
 A: *

*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)
A: Here are the things I use Grobner bases for, which I certainly find interesting:


*

*Extending the univariate division algorithm to multivariate polynomials (although not a true euclidean division algorithm, it is still useful).

*(related) Computing generators for $I_1 + I_2$ where $I_1,I_2$ are ideals in a multivariate polynomial ring (say $\mathbb{C}$), and using this to determine $I(V_1\cap V_2)$ where $V_1$ and $V_2$ are affine varieties in $\mathbb{A}^n$ for $n > 1$.
I'm not sure if these interest you or the students you are presenting to, but hopefully it's at least a start.
A: 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.
A: Since Gröbner basis algorithms may be considered as nonlinear generalizations of Gaussian elimination for systems of linear equations, they have very widespread applicability. Below is a random collection of applications of Gröbner bases.

*

*effective computation with (holonomic) special functions


*solving Diophantine equations (Pell)


*automated geometry theorem proving.


*coding theory


*signal and image processing


*robotics


*graph coloring problems e.g. Sudoku puzzles


*extrapolating "missing links" in palaeontology, and phylogenetic tree construction
