Applications of Gröbner bases

Question

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?

Answer 1

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

Answer 2

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

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

2. (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.

Answer 3

• 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)

Answer 4

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.