Digital Object Identifier (DOI) 10.1007/s00200-004-0158-4 AAECC 15, 233–266 (2004) Various New Expressions for Subresultants and Their Applications Gema M. Diaz–Toca 1, ∗ , Laureano Gonzalez–Vega 2,∗ 1 Dpto. de Matematica Aplicada, Universidad de Murcia, 30071 Murcia, Spain (e-mail: gemadiaz@um.es) 2 Dpto. de Matematicas, Estadistica y Comp., Universidad de Cantabria, Spain (e-mail: laureano.gonzalez@unican.es) Received: April 15, 2003; revised version: April 28, 2004 Published online: October 15, 2004 – © Springer-Verlag 2004 Abstract. This article is devoted to presenting new expressions for Subresul- tant Polynomials, written in terms of some minors of matrices different from the Sylvester matrix. Moreover, via these expressions, we provide new proofs for formulas which associate the Subresultant polynomials and the roots of the two polynomials. By one hand, we present a new proof for the formula intro- duced by J. J. Sylvester in 1839, formula written in terms of a single sum over the roots. By other hand, we introduce a new expression in terms of the roots by considering the Newton basis. Keywords: Subresultant polynomials and roots, Matrix theory, Bezout matrix Introduction One of the main tools in computer algebra to deal with polynomials in one variable is Subresultant polynomials. For example, they provide fraction free algorithms for computing the greatest common divisor of two polynomials, with a good behaviour under specialization. Their multiple properties over integral domains can be found in [8], [20], [21], [26], [27]. See [11] for extensions of the main results over integral domains to arbitrary commutative rings. They are also used in algorithms performing quantifier elimination or cylindrical algebraic decomposition (see [15] and [17]). In [13], an interesting historical discussion about Subresultant polynomials and polynomial remainder sequences is found. ∗ Partially supported by the European Union funded project RAAG (HPRN–CT–2001–00271) and by the spanish grant BFM2002-04402-C02-0