I'm trying to determine the smallest positive integer represented by a certain (positive, definite, binary) quadratic, and I came across this post: Minimum value of a positive definite binary quadratic form along integers which makes the claim that the smallest positive integer represented by a quadratic is the first coefficient of its reduced form. This kind of makes intuitive sense when I start looking at the reduced form, but I can't figure out how to prove it. Would someone mind explaining how to prove this result?

  • $\begingroup$ You do not give your form. Is it in two variables? $\endgroup$ – Will Jagy Mar 15 '18 at 17:12
  • $\begingroup$ Yes; sorry, I edited the question to reflect this. $\endgroup$ – Joe Mar 15 '18 at 17:15

If $F(x,y) = ax^2+bxy+cy^2$ is the positive definite reduced form, then first, it is clear that $F(x,y) \ge (a-|b|+c)\min(x^2,y^2)$ (just enumerate the cases). Now, if $xy\ne 0$, then $F(x,y)\ge a$ since $c-|b|\ge 0$. Further, if $a<c$, then $F(x,y) > a$.

Next, note that if $y=0$, then $F(x,y) = ax^2 \ge a$ unless $x=0$, and if $x=0$, then $F(x,y) = cy^2\ge c$ unless $c=0$. It follows that $F(x,y) \ge a$ unless $(x,y) = (0,0)$.

If $c\ne a$, then $f(x,y) > a$ when $xy\ne 0$, so $F$ can achieve its minimum value only when $x=0$ or $y=0$; since $a<c$, this can occur only when $x=\pm 1$.

So the minimum value is $a$, and if $a\ne c$, it is achieved only for $(x,y) = (\pm 1, 0)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.