Problems

There are a lot of problems here. Just do the ones that interest you.

1.

How many constants are needed in the general equation of a plane curve of degree n? (Check that your formula gives the right answer, 10, for the case n=3.)

2.

Let f(x)=x³+Ax+B where A and B are numbers. Let $\Delta=-(4A^3+27B^2)$ . Prove that

(a): f(x) has a multiple root if and only if $\Delta=0$ .
(b): f(x) has three distinct real roots if and only if $\Delta > 0$ .
(c): f(x) has one real root and two non-real roots if and only if $\Delta < 0$ .

(Hint: f(x) factors completely into linear factors over the complex numbers. Since there is no x² term in f(x), the sum of the zeros of f(x) is 0, and the factorization has the form

f(x) = (x-r)(x-s)(x+r+s)

for some complex numbers r and s. Calculate $\Delta$ in terms of r and sand factor it.)

The number $\Delta$ is called the discriminant; it plays a role analogous to that of b²-4ac for quadratic polynomials.

3.

It turns out that the real points on the elliptic curve y²=x³+Ax+B form two connected components if $\Delta > 0$ and only one connected component if $\Delta < 0$ . (Loosely speaking, a connected component is a piece you can draw without lifting your pencil from the paper.) Can you explain this, using the previous problem?

4.

Parameterize the rational points on the hyperbola x²-2y²=1.

5.

Parameterize the rational points on the sphere x²+y²+z²=1.

6.

(a) Prove that the circle x²+y²=3 has no rational points. (Hint: show that a rational point would give rise to a triple of integers (a,b,c) not all divisible by 3, such that a²+b²=3c². Examine the possibilities for a,b,c modulo 3.)

(b) Find some other integers n>0 such that x²+y²=n has no rational points.

7.

Let X be the curve y²=x³+x².

(a) Is X an elliptic curve?

(b) Draw a sketch of the curve X. The point P=(0,0), where two ``branches'' cross, is called a node, which is the simplest kind of singularity.

(c) Show that using lines of rational slope through the special point P yields a parameterization of the rational points on X. (You might need to exclude P and/or to exclude certain slopes.)

8.

Let E be the elliptic curve y²=x(x+5)(x-5)used in our examples. List all the rational points on E you know, and then calculate P+Q for some pairs of these to find more.

9.

Let E be an elliptic curve, and let P be a point on E other than O. Show that P+P=O if and only if the y-coordinate of P is zero. (This shows that in an elliptic curve, P+P=O does not imply P=O! One cannot divide by 2!)

10.

Find an elliptic curve with a rational point $P \not= O$ satisfying P+P+P=O. Hint: if a line L intersects E only at a single point P, and in particular does not pass through O (i.e., it is not vertical and is not the line at infinity), then by Bezout's Theorem, $L \cap E$ must be P with multiplicity 3, so P+P+P=0.

11.

Find eight rational points on the curve x⁴+y⁴=17.