Maria Francis

Reduced Grobner bases and Macaulay-Buchberger Basis Theorem over Noetherian rings

In this paper, we extend the characterization of Zx]/(f), where f is an element of Zx] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Grobner 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. Ax(1), ... , x(n)]/a, where a subset of Ax(1), ... , x(n)] is an ideal. We give some insights into the characterization for two special cases, when A = Z and A = ktheta(1), ... , theta(m)]. 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 ideal lattices, Gröbner bases and generalized hash functions

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Grobner bases. Univariate ideal lattices are ideals in the residue class ring, ℤ[x]/〈f〉 (here f is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Grobner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in ℤ[x1,…,xn], and use its computational hardness to establish the collision resistance of the hash functions.

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

Abstract Given an ideal a in A [ x 1 , … , x n ] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A [ x 1 , … , x n ] / a , when the residue class ring is a free A -module. When A is a field, the Krull dimension of A [ x 1 , … , x n ] / a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain A we introduce the notion of combinatorial dimension of A [ x 1 , … , x n ] / a and give a Grobner basis method to compute it for residue class rings that have a free A -module representation w.r.t. a lexicographic ordering. For such A -algebras, we derive a relation between Krull dimension and combinatorial dimension of A [ x 1 , … , x n ] / a . An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of A [ x 1 , … , x n ] / a without having to consider individual properties of the ideal. For A -algebras that have a free A -module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Grobner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A -algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A -algebras with a free A -module representation w.r.t. a degree compatible ordering as well.

On Gröbner bases over rings and residue class polynomial rings with torsion

Buchberger (1965) introduced Gröbner bases theory for polynomial rings over fields to give an algorithmic technique to determine a vector space basis of the residue class ring of a zero dimensional ideal. This is called as Macauley-Buchberger basis theorem, as Buchberger’s result is based on the work by Macaulay (1916). The Macaulay-Buchberger Basis theorem has been extended from polynomial rings over fields to Noetherian rings for free residue class rings in (Francis & Dukkipati, 2014). For this a Gröbner basis characterization has been given that led to an algorithmic method to test whether a residue class polynomial ring over any Noetherian rings is free or not (i.e there exists a module basis or not). This result is also an elegant generalization of the fact that Z[x]/〈f〉 is free if and only f is monic, to a mulivariate case, over any Noetherian rings. In this work (Dukkipati et al., 2014) we generalize the Macaulay-Buchberger basis theorem for residue class ring A[x1, . . . , xn]/a, in the case when it is finitely generated as an A-module but need not necessarily a free module, where A is a Noetherian ring and a is an ideal. This generalization gives us an insight into the nature of generating sets that span A-module A[x1, . . . , xn]/a and allows to study the concept of border bases over rings. We present a border division algorithm over rings and prove termination of the algorithm for a special class of border bases called acyclic border bases. We show the existence of such border bases and present some characterizations in this regard. We believe that this study helps in improving Gröbner basis methods for polynomial rings over rings in the cases, where ideals that give rise to (i) finitely generated residue class polynomial rings that are free, and (ii) finitely generated residue class polynomial rings with torsion.