> My GP-Pari program will solve for the roots, using some type of numerical > algorithm to hundreds of thousands of digits. But what I want to know, is > the very nature of this problem intrinsically involve a quartic so that no > matter how hard we try, it will always pop up?
I think the answer is "yes", even though I'm not quite sure exactly what your question means.
Let me say what my "yes" means. If you're given the lengths a and b of the two ladders, and the height h of their intersection, then the distance x between the two vertical walls is given by a quartic equation that is in general irreducible, although for particular values of a,b,h it might just happen to have a rational root.
It's easy to find such values: we can suppose the bottom of the picture looks like this:
\ / X /|\ 15/ | \13 / 12 \ |/___|___\| 9 5 for instance. Then obviously we have x = 14 and since the top half of the figure will look like:
|\ | \13u /| 12u| \ 5v/ |4v | \ / | | 5u X 3v|
we must have
a = 13u + 13, where 5u = 9 and b = 15 + 5v, where 3v = 5,
giving a = 182/5, b = 70/3, h = 12, (for x = 14).
I rescale this to get a simpler case: multiplying by 15/14 gives