Consider the set of all reducible integer-valued polynomials in $\mathbb Z$, meaning all those which can be factored into at least two integer-valued parts. My question is as to the sort of properties you could expect to see from this set, particularly as if they were treated as plotted on the Cartesian plane as usual.
I know they'd be countably infinite in number. My best guess is that for any given finite set of $(x,y)$ coordinates, you could find infinitely many of them that intersected all points in the set, even the prime $y$ coordinates. I do realize that any particular polynomial could intersect at most a finite number of prime $y$ values. Am I correct in my assumption that they would essentially saturate the plane?
And I have a follow-up question, particularly if I'm generally right about the above. Suppose you were to introduce an irreducible function into the mix, $x^2+1$ or the like. I would expect it to intersect with infinitely many other polynomials at every finite point it reaches, even the primes. Assuming that the notion of finite polynomial primes is a crock, it would reach infinitely many primes, but it still seems to me as though it would always be intersecting other functions there; its only real difference from the reducible functions would be in its ability to reach infinitely many more.
I apologize for not being able to state this more rigorously, but hopefully my general question is clear: is my understanding of how things would look more or less right (particularly w.r.t the saturation of all $(x,y)$ coordinates by the reducible polynomial functions), or have I erred somewhere?
Edit
To simplify things, here's my main question: is it true that infinitely many reducible polynomial functions will intersect every integral $(x,y)$ point?