The Dedekind-Mertens lemma and the contents of polynomials

Abstract

Abstract. Let R be a commutative ring, let X be an indeterminate, and let g ∈ R [X]. There has been much recent work concerned with determining the Dedekind-Mertens number}, especially on determining when µ R (g) = 1. In this note we introduce a universal Dedekind-Mertens number uµ R (g), which takes into account the fact that µ S (g) ≤ deg(g) + 1 for any ring S containing R as a subring, and show that uµ R (g) behaves more predictably than µ R (g).