Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this geometrically?
Asked
Active
Viewed 490 times
0
-
You had best typeset in the thing for which you want solutions. Also identify what particular theorem of Legendre is involved. – Will Jagy Jun 01 '16 at 23:47
-
The same equation reduces to Pell's equation. There solution is always there. http://math.stackexchange.com/questions/794510/curves-triangular-numbers – individ Jun 02 '16 at 12:34
1 Answers
1
A generic line intersects a conic in $2$ points.
If the conic and the line are rational (given by rational coefficients), then the set consisting of those two points is invariant by $\Bbb Q$-automorphisms, so either each point is rational, either they are defined over a quadratic extension of $\Bbb Q$ and the automorphism of that extension swaps the two intersection points.
If you have a rational point $P$ on the conic, then the rational lines going through it will intersect the rational conic at another rational point (possibly $P$ itself if the line is tangent at $P$ to the conic)
Moreover, two different lines going through $P$ can only intersect at $P$, so the lines that are not tangent to the conic at $P$ give you distinct points on the conic.
This actually gives you a birational map between $\Bbb P^1$ and the conic.
mercio
- 51,119