Automated Practical Reasoning

Part of the series Texts and Monographs in Symbolic Computation pp 7-37

An Algorithm for Solving Systems of Algebraic Equations in Three Variables

  • Michael Kalkbrener

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The most famous algorithm for computing the greatest common divisor (gcd) of two univariate polynomials over a field is, undoubtedly, the algorithm of Euclid. In attempting to generalize it to the multivariate case one easily arrives at the concept of polynomial remainder sequences and discovers the phenomenon of explosive coefficient growth (see, e.g., Brown 1971). To overcome this problem primitive polynomial remainder sequences (pprs) have been introduced. However, the classical primitive polynomial remainder sequence algorithm for computing gcds (Brown 1971) has one rather obvious disadvantage. In order to make a polynomial primitive, its content, which is the gcd of its coefficients, has to be computed. Therefore, additional gcd computations in the coefficient domain are necessary for computing a pprs. Fortunately, the costs of these content computations can be considerably reduced by subresultant techniques (Collins 1967, Brown and Traub 1971, Brown 1978) or trial division (Hearn 1979, Stoutemyer 1985).