Compositions of Polynomials with Coefficients in a Given Field

Let ⊂ be fields of characteristic 0, and let [] denote the ring of polynomials with coefficients in . Let , ≠ 0. For ∈ []\[], define (), the deficit of , to equal − max{0 ≤ ≤ : ∉ }. For ∈ [], define () = . Let and let , with ≠ 0, ≠ 0, , ∈ , ∉ for some ≥ 1. Suppose that ∈ [], ∈ []\[], not constant. Our main result is that ∘ ∉ [] and ( ∘ ) = (). With only the assumption that ∈ , we prove the inequality ( ∘ ) ≥ (). This inequality also holds if and are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of . Finally we extend our results to polynomials in two variables and compositions of the form ((, )), where is a polynomial in one variable