Generalized Polynomial Remainder Sequences

Given two polynomials over an integral domain, the problem is to compute their polynomial remainder sequence (p.r.s.) over the same domain. Following Habicht, we show how certain powers of leading coefficients enter systematically all following remainders. The key tool is the subresultant chain of two polynomials. We study the primitive, the reduced and the improved subresultant p.r.s. algorithm of Brown and Collins as basis for Computing polynomial greatest common divisors, resultants or Sturm sequences. Habicht’s subresultant theorem allows new and simple proofs of many results and algorithms found in different ways in Computer algebra.