Compositions of polynomials with coefficients in a given field

Let F subset of K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = Sigma(k=0)(n) a(k)x(k) is an element of K[x], a(n) not equal 0. For P is an element of K[x]\F[x], define D-F(p), the F deficit of p, to equal n - max{0 less than or equal to k less than or equal to n : a(k) is not an element of F}. For p is an element of F[x], define D-F(p) = n. Let p(x) = Sigma(k=0)(n) a(k)x(k) and let q(x) = Sigma(j=0)(m) b(j)x(j) with a(n) not equal 0, b(m) not equal 0, a(n), b(m) is an element of F, b(j) is not an element of F for some j greater than or equal to 1. Suppose that p is an element of K[x], q is an element of K[x]\F[x], p not constant. Our main result is that p circle q is not an element of F[x] and D-F(p circle q) = D-F(q). With only the assumption that a(n)b(m) is an element of F, we prove the inequality D-F(p circle q) greater than or equal to D-F(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. (C) 2002 Elsevier Science (USA).