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

  • Authors:

  • Äbdelilah Kandri-Rody;Deepak Kapur

  • Affiliations:

  • Department des Mathematiques Faculte des Sciences University Cadi Ayyad Marrakech, Morocco;Computer Science Branch General Electric Company Corporate Research and Development Schenectady, New York 12345, USA

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 1988

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Abstract

An algorithm for computing a Grobner 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 Grobner bases of polynomial ideals over an arbitrary field and over the integers as well as our algorithms for computing Grobner 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 Grobner 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 Grobner basis of an ideal, determined by an admissible ordering on terms (power products) and other conditions, is discussed.