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 $f(x)=x^3+Ax+B$ where $A$ and $B$ are numbers. Let $\Delta=-(4A^3+27B^2)$. Prove that
   1. $f(x)$ has a multiple root if and only if $\Delta=0$.
   2. $f(x)$ has three distinct real roots if and only if $\Delta > 0$.
   3. $f(x)$ 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 $x^2$ 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 $s$ and factor it.)

   The number $\Delta$ is called the discriminant; it plays a role analogous to that of $b^2-4ac$ for quadratic polynomials.

3. It turns out that the real points on the elliptic curve $y^2 = x^3 + Ax + B$ 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 $x^2-2y^2=1$.

5. Parameterize the rational points on the sphere $x^2+y^2+z^2=1$.

6. (a) Prove that the circle $x^2+y^2=3$ 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 $a^2+b^2=3c^2$. Examine the possibilities for $a,b,c$ modulo $3$.)

   (b) Find some other integers $n>0$ such that $x^2+y^2=n$ has no rational points.

7. Let $X$ be the curve $y^2=x^3+x^2$.

   (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 exclude certain slopes.)

8. Let $E$ be the elliptic curve $y^2=x(x+5)(x-5)$ 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 $x^4+y^4=17$.

©Berkeley Math Circle