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I guess this question might be very difficult, so I would only like to know what is known about it.

Given a quadratic polynomial $P$ in two variables with degree $2$ and integer coefficients. Assume $P = 0$ has real solutions. Does this imply that there are also rational solutions to it? I am really only interested in quadratic polynomials. In case the answer is yes: are the rational solutions dense? In case of no: Is it a 'common' or 'rare' event to have a polynom without rational solutions?

thanks Till

Till
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1 Answers1

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The quadratic equation $x^2+y^2-3=0$ has real but no rational solutions.

If a non-singular conic $C$ defined over $\mathbb{Q}$ has a rational point $P$ then all rational points on $P$ can be parameterised. Let $t\in\mathbb{Q}$ and let $L_t$ be the line through $P$ with slope $t$. Then $L_t$ meets $C$ in two points: $P$ and $Q_t$. Then $t\mapsto Q_t$ is a parameterisation of the rational points on $C$. (OK you need to also consider the "$t=\infty$" case).

From the parameterisation, it is easily seen that the rational points are dense within the real points.

As a classic example, let $C$ be the unit circle $x^2+y^2-1$ and $P$ the point $(1,0)$. The line $L_t$ has equation $y=t(x-1)$ which meets $C$ in $P$ and $Q_t=((t^2-1)/(1+t^2),-2t/(1+t^2))$. As $t$ varies over $\Bbb Q$ we all the rational points on the unit circle. To see they are dense in the real points, a typical real point is also $Q_t$ but with $t\in\Bbb R$. Just take a sequence of rationals converging to $t$.

Angina Seng
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