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?

See http://www.msri.org/people/members/chillar/files/gbapplfinal.pdf — lhf, Apr 12 '11 at 01:09
Outside mathematics, the actual industrial application is in robotics. — , Jan 16 '12 at 02:26

score 21 · Accepted Answer · edited Apr 28 '22 at 19:04

21

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

edited Apr 28 '22 at 19:04

The Amplitwist

1,335

answered Apr 12 '11 at 00:50

Bill Dubuque

282,220

I was unable to fix the last link over "phylogenetic tree construction", which points to springerlink.com. Perhaps you could take a look, whenever possible... – The Amplitwist Apr 28 '22 at 19:06

score 7 · Answer 2 · edited Jun 12 '20 at 10:38

7

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.

edited Jun 12 '20 at 10:38

Community

1

answered Apr 12 '11 at 02:55

Somewhat similar... – J. M. ain't a mathematician Apr 12 '11 at 02:59

score 6 · Answer 3 · answered Apr 12 '11 at 00:29

6

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.

answered Apr 12 '11 at 00:29

Alex Becker

61,883

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I don't think you need Gröbner bases to find generators for $I_1+I_2$ -- any generators of $I_1$ and of $I_2$ can be combined to give generators of $I_1+I_2$. A slightly different answer that actually uses Gröbner bases is that Gröbner bases give a way to compute generators of the intersection $I_1 \cap I_2$. (Briefly, $I_1 \cap I_2 = ((1-t)I_1 + tI_2) \cap k[x]$, and this intersection, eliminating the extra variable $t$, can be computed with Gröbner bases in an elimination order.) – Zach Teitler Nov 10 '21 at 07:52

score 6 · Answer 4 · answered Apr 12 '11 at 02:45

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)

Applications of Gröbner bases

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