Gaussian Polynomials and Content Ideals

We prove that every regular Gaussian polynomial over a locally Noetherian ring has invertible content ideal. We do this by first proving that Gaussian polynomials over an approximately Gorenstein local ring have principal content ideal. We also show over locally Noetherian rings that a regular polynomial f of degree n is Gaussian if c(fg) = c(f)c(g) for polynomials g of degree at most n. Introduction A question which appears to have originated in the 1965 thesis of Kaplansky’s student Tsang asks whether over an integral domain the content of a nonzero Gaussian polynomial is an invertible ideal. Let R be a commutative ring. The content c(f) of a polynomial f ∈ R[t] is the ideal of R generated by the coefficients of f . A polynomial f(t) ∈ R[t] is Gaussian over R if for every polynomial g(t) ∈ R[t] we have c(fg) = c(f)c(g). A recent paper of Glaz and Vasconcelos [GV] on Gaussian polynomials and content ideals has motivated our interest in this topic. Glaz and Vasconcelos prove that a nonzero Gaussian polynomial over a normal Noetherian domain has invertible content ideal [GV, Theorem 4.4]. They also prove that a Gaussian polynomial over an integrally closed quasilocal domain (R,m) has principal content ideal provided the residue field R/m has characteristic p > 0 [GV, Theorem 3.1]. We are able to prove that rings that are locally approximately Gorenstein also have this property: a regular Gaussian polynomial over such a ring has invertible content ideal. See Corollary 2.7. Using this we prove that Gaussian polynomials over a locally Noetherian integral domain satisfy the conjecture of Tsang-Glaz-Vasconcelos (cf. Corollary 3.4). As an extension of the question from Tsang’s thesis, it is asked in [AK, Question 1.4] whether over an integral domain R a polynomial f(t) ∈ R[t] of degree n has an invertible content ideal if c(fg) = c(f)c(g) for all polynomials g(t) ∈ R[t] of degree at most n. We show in Corollary 3.4 that all Noetherian integral domains have this property. Let us say a polynomial f ∈ R[t] is Gaussian for polynomials of degree at most m if c(fg) = c(f)c(g) for all g ∈ R[t] with deg(g) ≤ m. If R is locally Noetherian and f(t) ∈ R[t] is of degree n and is regular, we show in Received by the editors October 18, 1995. 1991 Mathematics Subject Classification. Primary 13A15, 13B25, 13G05, 13H10. 1This question is explicitly given as Conjecture (1.1) in the paper of Sarah Glaz and Wolmer Vasconcelos [GV]. c ©1997 American Mathematical Society