For a quadratic endomorphism of the affine line defined over the rationals we
consider the problem of bounding the number of rational points that eventually
land at a given constant after iteration, called pre-images of the constant. In
the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,"
it was shown that the number of rational pre-images is bounded as one varies
the morphism in a certain one-dimensional family. Explicit values of the
constant for pre-images of zero and -1 defined over the rational numbers were
addressed in subsequent articles. This article addresses an explicit bound for
any algebraic image constant and provides insight into the geometry of the
"pre-image surfaces."Comment: to appear in Involve; 16page
