Compositions of Polynomials with Coefficients in a given Field

Abstract

Let F and K be fields of characteristic 0, with F a subset of K. Let K[x] denote the ring of polynomials with coefficients in K. For p in K[x]\F[x], deg(p) = n, let r be the highest power of x with a coefficient not in F. We define the F deficit of p to be D_F(p) = n-r. For p in F[x], D_F(p) = n. Suppose that the leading coeffcients of p and q are in F, and that some coefficient of q(other than the constant term) is not in F. Our main result is that the F deficit of the composition of p with q equals the F deficit of q. This implies our earlier result: If p(q(x)) is in F[x]then p is in F[x] and/or q is in F[x]. We also prove similar results for compositions of the form p(q(x,y)), for the iterates of a polynomial, and for fields of finite characteristic, if the characteristic of the field does not divide the degree of p. Finally, If F and K are only rings, then we prove the inequality D_F(p(q(x))) >=D_F(q).