Norms of Factors of Polynomials

for any positive integer k is bounded above by a quantity that is independent of k. Hence, whenever A(x) is divisible by a cylcotomic polynomial and N is su ciently large, there will be Q(x) 2 Z[x] with arbitrarily large Euclidean norm and with kAQk N . It is reasonable, however, to expect that the Euclidean norm of Q(x) is bounded whenever A(x) is free of cyclotomic factors. This in fact is the main result of this paper. Theorem 1. Let A(x) 2 Z[x] be a polynomial having no cyclotomic factors. Let N 1. If Q(x) 2 Z[x] and kA(x)Q(x)k N , then kQk is bounded by a function depending only on A(x) and N . The bound on kQk can be made explicit, and this will be clear from the arguments. There are special cases where such a bound follows from the