Content Formulas for Polynomials and Power Series and Complete Integral Closure

Gaussian polynomials and invertibility

“…Previously, the bound for m was known to be less than or equal to the number of nonzero coefficients of g. If c(f ) is an invertible ideal of R, multiplying both sides by the appropriate power of the inverse of c(f ) reduces the formula to c(f g) = c(f )c(g) no matter the degree of g. More generally, if c(f )R M is principal for each maximal ideal M (but perhaps with a nonzero annihilator for some or all M ), then Nakayama's Lemma can be used to show that c(f )c(g)R M = c(f g)R M for each M (or simply apply the aforementioned [HH2,Theorem 2.1] with the roles of f and g reversed). The conclusion is then reached by noting that an ideal is completely characterized by its localizations at maximal ideals (see, for example, [A], or [AK,Theorem 1.1…”

The content of a Gaussian polynomial is invertible

“…[1,7]). However, if u and v are indeterminates over R, then the polynomial s + tX is not Gaussian over the ring extension R [u, v]…”

The Dedekind-Mertens lemma and the contents of polynomials

“…In [10], [4], the question of the opposite inequality to (0.2) was also considered. The special case of whether µ R (g) = 1 implies c R (g) is principal was considered as early as [14], and several further results have recently been obtained on this case ( [2], [3], [4], [5], [7], [8], [9], [10]). …”

“…Many papers ( [1], [2], [3], [4], [5], [7], [8], [9], [10], [13]) have recently considered questions concerning the following well-known result which is usually called the Dedekind-Mertens Lemma:…”