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Publication Details
AFRICAN RESEARCH NEXUS
SHINING A SPOTLIGHT ON AFRICAN RESEARCH
mathematics
Computing a Gröbner basis of a polynomial ideal over a Euclidean domain
Journal of Symbolic Computation, Volume 6, No. 1, Year 1988
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Description
An 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. © 1988, Academic Press Limited. All rights reserved.
Authors & Co-Authors
Kandri Rody, Abdelilah
Morocco, Marakech
Université Cadi Ayyad
Kapur, Deepak
United States, Niskayuna
Ge Global Research
Statistics
Citations: 84
Authors: 2
Affiliations: 2
Identifiers
Doi:
10.1016/S0747-7171(88)80020-8
ISSN:
07477171