This is great to include such problems. Here are a few examples of nice doable problems:
- 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.
- 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 $a=b$.
- For the given ellipsoid
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.