Counting Reducible Composites of Polynomials

This research answers some open questions about the number of reducible translates of a fixed non-constant polynomial over a field. The natural hypothesis to consider is that the base field is algebraically closed in the function field. Since two possible choices for the base field arise, this naturally yields two different hypotheses. In this work, we explicitly relate the two hypotheses arising from this choice. Using the theory of derivations, and specifically an explicit construction of a derivation with a well-understood ring of constants, we can relate the ranks of the two relative-unit-groups involved, both of which are free Abelian groups under our hypothesis. These results allow us to give a more natural (though not stronger) bound on the number of reducible translates than the bound that was previously known. Also, we extend this count of reducible translates to a count of reducible composites and find that a similar bound will hold in this more general setting.