Bounds on factors in Z[x]

We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few improvements to some existing bounds. We compare these bounds, and for each bound give explicit examples where that bound is the best; thus showing that no one bound is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have “unexpectedly” large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins.