# 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 ? (Check that your formula gives the right answer, 10, for the case .)

2. Let where and are numbers. Let . Prove that
1. has a multiple root if and only if .
2. has three distinct real roots if and only if .
3. has one real root and two non-real roots if and only if .
(Hint: factors completely into linear factors over the complex numbers. Since there is no term in , the sum of the zeros of is 0, and the factorization has the form

for some complex numbers and . Calculate in terms of and and factor it.)

The number is called the discriminant; it plays a role analogous to that of for quadratic polynomials.

3. It turns out that the real points on the elliptic curve form two connected components if and only one connected component if . (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 .

5. Parameterize the rational points on the sphere .

6. (a) Prove that the circle has no rational points. (Hint: show that a rational point would give rise to a triple of integers not all divisible by 3, such that . Examine the possibilities for modulo .)

(b) Find some other integers such that has no rational points.

7. Let be the curve .

(a) Is an elliptic curve?

(b) Draw a sketch of the curve . The point , 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 yields a parameterization of the rational points on . (You might need to exclude and/or exclude certain slopes.)

8. Let be the elliptic curve used in our examples. List all the rational points on you know, and then calculate for some pairs of these to find more.

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

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

11. Find eight rational points on the curve .