Leírás
Suppose one wants to count points $(x,y)$ on a line $ax+by=c$, in which $x=u/w$ and $y=v/w$ are rational numbers. One might require that $u,v,w$ are integers with no common factor, and count solutions with $|u|,|v|,|w|\le B$. How does this number grow as $B$ tends to infinity? This question is fairly easy and "classical". It turns out that the counting function is roughly $C_{a,b,c}B$, for some positive constant $C_{a,b,c}$. However to see this behaviour, $B$ must be large enough compared to the coefficients $a,b,c$; and it turns out that the point at which the correct asymptotic behaviour begins will depend on the size of the smallest solution to the original equation $ax+by=c$. All this will be described in the talk, which will go on to discuss the analogous situation for conics.