A binary quadratic form whose range topograph has a lake

Question

Is it true that if Q is a binary quadratic form whose range topograph has a lake, then Q factors as a product of two linear forms with integer coefficients?

Answer 1

Alright, Weissman's book, which I have. Very nice.

On page 274, definition 10.21 says a lake is a place where the value of the form is zero. I don't see it, but any vector in the "domain topograph" is nonzero, at least one of the elements of $(x,y)$ is nonzero. This follows from the fact that we extend from the "home" basis. Any lax basis, a pair of vectors with $\pm$ signs, gives determinant $\pm 1$ for the little 2 by 2 matrix. See note upper right on page 233.

That is about all we need. A lake says we have integers $u,v$ not both zero, with $$ au^2 + b uv + c v^2 = 0 \; . \; $$

Now, if $v=0,$ we know that $u \neq 0,$ so $a=0.$ In this special case, we can factor $bxy + c y^2$ as $(bx+cy)y.$

If $v \neq 0,$ we can divide through by $v^2, $ then define rational $r = \frac{u}{v},$ so that $$ a r^2 + b r + c = 0 . $$ As we have a rational root, the Quadratic Formula says that the traditional discriminant $$ \Delta = b^2 - 4ac $$ must be an integer square, $\Delta = \delta^2.$ Then we can factor $ax^2 + bxy + c y^2$ over the integers, I give a method and proof at Prove that if $b^2-4ac=k^2$ then $ax^2+bx+c$ is factorizable