When and how do we dehomogenize a homogeneous function?
To solve Prove the sign and zeroes of $Ax^2 + 2Bxy + Cy^2$ (without using the second derivative test) , "user" set $$t = \frac x y$$ and wrote $$Ax^2 + 2Bxy + Cy^2 =y^2\left(At^2+2Bt+C\right)$$ to solve the problem.
Where did that come from? Ted Shifrin wrote
When you study a homogeneous function, it is natural to dehomogenize by evaluating $f(x,y)$ at either $(1,y/x)$ or $(x/y,1)$
I'd like to learn more about this technique, and am unable to find a reference here or via Google. What does it mean to dehomogenize a function? When and how do we do it?
Update
Although I'm still looking for a reference, I worked out a "derivation" in this case, to which I ask for feedback and verification:
Goal: Characterize the sign of $$P(x,y) = Ax^2 + 2Bxy + Cy^2$$ on $\mathbb R^2 - \{(0,0)\}$.
Approach: Before attempting any algebra, understand $P$ qualitatively. Two things are clear:
(i) When $|x|$ is large, $A$ dominates, and when $|y|$ is large, $C$ dominates; it's only when $|x| \approx |y|$ that things are messy
(ii) The absolute magnitude of $x$ and $y$ are irrelevant; it's only their ratio that matters.
This suggests that we divide by $x^2$, which will preserve the sign (which is what we care about) and perhaps replace absolute magnitude with ratio. (We can handle $x = 0$ later.)
$$P(x,y) = {x^2} (A + B \frac y x + C \frac {y^2} {x^2}) \\ \operatorname{sgn}[P(x,y)] = \operatorname{sgn}(A + B \frac y x + C \frac {y^2} {x^2})$$ which is a polynomial in $\frac y x$, whose range of signs is obvious from its discriminant.
Furthermore, we can assume WLOG that $x \neq 0$, since if $x = 0$, then $y \neq 0$, and $\operatorname{sgn} P$ is the same by symmetry.
