Requesting a Polynomial System of Equations

Question

I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class.

I usually give my students some project proposals which they can choose from, in case they cannot come up with an interesting problem statement themselves. I try to be as varied as possible in the problems I propose, covering most areas of commutative algebra.

One such area I wanted to include is Gröbner bases and Elimination theory. I wanted to ask the Mathematics StackExchange community if they knew of any interesting polynomial systems of equations, perhaps arising from physics or chemistry, that are not completely straightforward but are also approachable for a graduate student of mathematics.

Answer 1

Here's a really nice textbook chapter that covers using algebraic geometry to compute equilibria in economics. It looks pretty approachable for students while simultaneously getting into plenty of nontrivial stuff. There are also exercises at the end of the chapter, which may serve as inspiration for project ideas.

Sturmfels, B. (2002). Polynomial Systems in Economics. In Solving systems of polynomial equations (pp. 235-246). American Mathematical Soc.

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Here's another pedagogical article, this time from robotics:

Wampler, C. W., & Sommese, A. J. (2011). Numerical algebraic geometry and algebraic kinematics. Acta Numerica, 20, 469-567.

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And here's a research paper that uses algebraic geometry to compute protein structures. Not as friendly as the above, but potentially more interesting in terms of project ideas.

Wang, L., Mettu, R. R., & Donald, B. R. (2005, August). An algebraic geometry approach to protein structure determination from NMR data. In 2005 IEEE Computational Systems Bioinformatics Conference (CSB'05) (pp. 235-246). IEEE.

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Answer 2

Boolean networks (i.e. $F_2$ valued systems) are a popular tool in mathematical biology for analysing gene regulatory networks. Fixed points correspond (roughly) to phenotypes and can be computed using Gröbner bases.

Here is a database of examples: https://booleangenenetworks.math.iastate.edu/

Answer 3

Two resources:

(1) A really nice website Gröbner bases and their applications. Includes many elementary examples coded in a Python package.

(2) Automatic theorem discovery in elementary geometry. The paper below applies this to several examples, one of which characterizes when a triangle's orthic triangle is isosceles.

Montes, Antonio, and Tomás Recio. "Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems." In Automated Deduction in Geometry: 6th International Workshop, ADG 2006, Pontevedra, Spain, August 31-September 2, 2006. Revised Papers 6, pp. 113-138. Springer Berlin Heidelberg, 2007.

Answer 4

This is great to include such problems. Here are a few examples of nice doable problems:

1. Any problem that involves Lagrange multipliers, a constraint curve given as a multivariate polynomial, and a function to be maximised/minimised also given as such polynomial.

Comment: If one has trig functions, e.g. sine and cosine, one can always replace them with new indeterminates say $X$ and $Y$ provided this extra polynomial $X^2+Y^2-1$ is added to the list of polynomials that give a basis in some ideal $I$ for which later Gröbner basis will be computed.

2. Let's ask this question: let's take an ellipse (or any other quadratic like a parabola or a hyperbola) say:

$x^2/a^2 + y^2/b^2 = 1$

with $a \ne b$ and let $r>0$ be a parameter. Imagine a circle of radius $r$ whose center $C$ is located on the ellipse at a point $(x_0,y_0)$. Now move the circle along the ellipse so that its center $C$ remains on the ellipse. At each point $C$ draw a line through $C$ perpendicular to the tangent at $C$ to the ellipse. Each such line intersects the circle at $C$ of radius $r$ at two points $P_1(x,y)$ and $P_2(x,y)$. Find two polynomials $p_1$ and $p_2$ in $x, y, a, b, x_0, y_0$ and $r$ such that the equations $p_1=0$ and $p_2=0$ give two curves traced by the points $P_1$ and $P_2$ when the center $C$ of the circle traces the ellipse. These curves will be equidistant to the ellipse at the distance $r$. Analyse the shape of these two curves for the given $a$ and $b$ as parameterised by $r$. For small enough $r < r_0$ for some $r_0$ versus the values of $a$ and $b$, the two curves will be smooth. When $r=r_0$, two singular point will appear. When $r>r_0$, each singular point will become a dovetail with three singular points. Find a formula for $r_0$ in terms of $a$ and $b$. Relate it to the curvature of the ellipse.

In another words, find an envelope of the ellipse for each $r>0$. What happens in the special case when $a=b$? Geometrically and algebraically?

Problems like these two above require a CAS like Maple, for example, to compute Gröbner bases. The way I have formulated problem 2 is a hint how to find the two polynomial curves for the ellipse or for any other quadratic. Nothing else is needed except the understanding of a tangent line and a normal line to a curve in a plane.

Warning: The polynomial curves equidistant do a true ellipse ($a \ne b$) are not ellipses (of course). It is easy to predict what they are when $a=b$.

3. For the given ellipsoid

$x^2/a^2+y^2/b^2+z^2/c^2=1$

and a point $P(x_0,y_0,z_0)$ not on the ellipsoid, find:

(a) a point $S$ on the ellipsoid closest to $P$.

(b) points $S_1$ and $S_2$ on the ellipsoid such that the lines $PS_1$ and $PS_2$ are normal to the ellipsoid.

As far as a reference, I recommend the book Ideals, Varieties, and Algorithms by Cox, Little & O'Shea on elementary Algebraic Geometry. It is a fantastic book whose initial chapters through Gröbner bases and the theory of elimination should be accessible to algebraically advanced high school students.

Good luck!

Rafal Ablamowicz [email protected]

Answer 5

Code theory, from the book "A course in error correcting codes", from Justesen and Hohold. Chapter 5 and 6 tell you how to relate a linear codes, Reed Solomon and cyclic codes, with special polynomials. It is enough the first 2 sections of those chapters and chapter 1, fist 2 sections, too.