Reduced Gröbner bases and Macaulay–Buchberger Basis Theorem over Noetherian rings

Abstract

In this paper, we extend the characterization of Z[x]/ f , where f ∈ Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay-Buchberger Basis Theorem) to rings, i.e. A[x1, . . . , xn]/a, where a ⊆ A[x1, . . . , xn] is an ideal. We give some insights into the characterization for two special cases, when A = Z and A = k[θ1, . . . , θm]. As an application of this characterization, we show that the concept of border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module.

On Gröbner bases and Krull dimension of residue class rings of polynomial rings over integral domains

“…, x n ]/a to have a free A-module representation w.r.t. a monomial order has been studied in (Francis & Dukkipati, 2014). Here, we show that this characterization can be extended to A[x 1 , .…”

“…G (or w.r.t. ≺) is given in (Francis & Dukkipati, 2014). One can easily extend this to residue class rings that are not finitely generated as shown below.…”

“…G (or w.r.t. ≺) makes use of the the concept of 'short reduced Gröbner basis' introduced in (Francis & Dukkipati, 2014) which we briefly describe below. Definition 3.1.…”

Border Bases for Polynomial Rings over Noetherian Rings

“…, x n ]/a if it is finitely generated (Francis & Dukkipati, 2014). We describe this briefly below and for more details one can refer to (Francis & Dukkipati, 2014). The notation is mostly borrowed from (Adams & Loustaunau, 1994).…”

“…The concept of border bases can be easily extended to polynomial rings over rings if the corresponding residue class ring has a free A-module representation w.r.t. some monomial order and is finitely generated as an A-module (Francis & Dukkipati, 2014). In this paper, we study border bases for ideals in polynomial rings over Noetherian commutative rings in a more general set up.…”