Computing a Gröbner basis of a polynomial ideal over a Euclidean domain

AbstractAn algorithm for computing a Gröbner basis of a polynomial ideal over a Euclidean domain is presented. The algorithm takes an ideal specified by a finite set of polynomials as its input; it produces another finite basis of the same ideal with the properties that using this basis, every polynomial in the ideal reduces to 0 and every polynomial in the polynomial ring reduces to a unique normal form. The algorithm is an extension of Buchberger's algorithms for computing Gröbner bases of polynomial ideals over an arbitrary field and over the integers as well as our algorithms for computing Gröbner bases of polynomial ideals over the integers and the Gaussian integers. The algorithm is simpler than other algorithms for polynomial ideals over a Euclidean domain reported in the literature; it is based on a natural way of simplifying polynomials by another polynomial using Euclid's division algorithm on the coefficients in polynomials. The algorithm is illustrated by showing how to compute Gröbner bases for polynomial ideals over the integers, the Gaussian integers as well as over algebraic integers in quadratic number fields admitting a division algorithm. A general theorem exhibiting the uniqueness of a reduced Gröbner basis of an ideal, determined by an admissible ordering on terms (power products) and other conditions, is discussed

Function definitions in term rewriting and applicative programming

The frameworks of unconditional and conditional Term Rewriting and Applicative systems are explored with the objective of using them for defining functions. In particular, a new operational semantics, Tue-Reduction, is elaborated for conditional term rewriting systems. For each framework, the concept of evaluation of terms invoking defined functions is formalized. We then discuss how it may be ensured that a function definition in each of these frameworks is meaningful, by defining restrictions that may be imposed to guarantee termination, unambiguity, and completeness of definition. The three frameworks are then compared, studying when a definition may be translated from one formalism to another

Complete Sets of Reductions Modulo A Class of Equational Theories which Generate Infinite Congruence Classes

In this paper we present a generalization of the Knuth-Bendix procedure for generating a complete set of reductions modulo an equational theory. Previous such completion procedures have been restricted to equational theories which generate finite congruence classes. The distinguishing feature of this work is that we are able to generate complete sets of reductions for some equational theories which generate infinite congruence classes. In particular, we are able to handle the class of equational theories which contain the associative, commutative, and identity laws for one or more operators. We first generalize the notion of rewriting modulo an equational theory to include a special form of conditional reduction. We are able to show that this conditional rewriting relation restores the finite termination property which is often lost when rewriting in the presence of infinite congruence classes. We then develop Church-Rosser tests based on the conditional rewriting relation and set forth a completion procedure incorporating these tests. Finally, we describe a computer program which implements the theory and give the results of several experiments using the program