Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$.
Is there a way of computing the integer points on this curve. Since it is affine an not projective we can't just find the rational points and clear denominators.
Thanks
On
April 2017. There has been some interest for the case with some coefficients $0.$ I decided to write it with $c=0.$
IF $c=0,$ so that $$a x^2 + b x y + d x + e y + f = 0, $$
THEN $$ ( a b x + b^2 y + b d - a e ) ( b x + e) = - b^2 f - a e^2 + bde. $$ The right hand side is an integer constant, all its divisors, positive and negative, can be found. For each divisor $$ t | (- b^2 f - a e^2 + bde), $$ solve $$ bx + e = t, $$ $$ a b x + b^2 y + b d - a e = \frac{- b^2 f - a e^2 + bde}{t}, $$ and make sure that both $x,y$ come out as integers.
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IF we take instead $a=0,$ so that $$ b x y + c y^2+ d x + e y + f = 0, $$
THEN $$ ( b^2 x + b c y + b e - cd ) ( b y + d) = - b^2 f - c d^2 + bde. $$
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IF we have both $a=c=0,$ $$ b x y + d x + e y + f = 0, $$
THEN $$ ( bx+e ) ( b y + d) = - b f + de, $$ where we see we got to erase one factor of $b$ throughout.
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April 2017: if, in the letters named in the question, we have $$ a=c=0, $$ see, among many such questions, How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?
Oh, forgot, as Qiaochu says, integers.
Took me a while; never bothered to write out the general case before, but it came out alright. In any case, you can check what I wrote against a multiple of what you wrote, see if i got it all correct.
Define $$ \Delta = b^2 - 4 a c $$
Then you are solving $$ \left( \Delta y + bd -2ae\right)^2 - \Delta \left(2ax+by+d \right)^2 = \left(bd -2ae \right)^2 - \Delta \left(d^2 - 4 a f \right) $$ where the final quantity, $ \left(d^2 - 4 a f \right),$ is not squared. As you can see, there is a solution when $f=0$ with $x,y = 0.$
If $ \Delta = b^2 - 4 a c $ is negative, there are, at most, finitely many solutions. If $ \Delta$ is zero or a positive square, the left hand side factors and there are finitely many solutions, if any. If $ \Delta$ is positive and not a square, there is a Pell type equation, if there are any solutions there are infinitely many. Finding all of them is a mess unless the right hand side has very small absolute value. Even then you need that subset of the Pell-like solutions that allow integer values of $x,y.$