Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = n ∑ k=0 akx k ∈ K[x], an 6= 0. For p ∈ K[x]\F [x], define DF (p), the F deficit of p, to equal n −max{0 ≤ k ≤ n : ak / ∈ F}. For p ∈ F [x], define DF (p) = n. Let p(x) = n ∑ k=0 akx k, q(x) = m ∑ j=0 bjx j , with an 6= 0, bm 6= 0, an, bm ∈ F , bj / ∈ F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F [x], p not constant. Our main result is that p ◦ q / ∈ F [x] and DF (p ◦ q) = DF (q). With only the assumption that anbm ∈ F, we prove the inequality DF (p ◦ q) ≥ DF (q). This inequality also holds if F and K 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 p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.