Joachim von zur Gathen,
Fachbereich 17 Mathematik-Informatik,Universität Paderborn
Paderborn, Germany.

Subresultants revisited

Starting in the late 1960s, Collins and Brown & Traub invented polynomial remainder sequences (PRS) in order to apply the Euclidean algorithm to integer polynomials. Subresultants play a major role in this theory. We compare the various notions of subresultants, give a general and precise definition of PRS, and clean up some loose ends:
  • prove a 1971 conjecture of Brown that all results in the subresultant PRS are integer polynomials,
  • show an exponential lower bound on the pseudo PRS.

Lastly, we show how Kronecker had, already in the 1870s, discovered many of the fundamental properties of Euclid's algorithm for polynomials.