Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$.
For $k = 1$, it can be done very fast. The Rational Root Theorem gives a set of candidates. But what for $k=2$? How can I split a polynomial into factors in this case?
Example: $$ax^2+bx+cy^2+dy+g=0$$ where $a,b,c,d,g$ are constants and $x,y$ are variables. This equation has integer solutions if and only if the quadratic equation $$ax^2+bx+(cy^2+dy+g)=0$$ has integer solutions with respect to $x$. For this case, we can use The Rational Root Theorem for the one variable $x$.
My question: Is this approach valid.