text: Subresultants In the case of univariate polynomials, there is a strong relationship between the greatest common divisors and resultants. More precisely, the resultant of two polynomials P, Q is a polynomial function of the coefficients of P and Q which has the value zero if and only if the GCD of P and Q is not constant.
  The subresultants theory is a generalization of this property that allows characterizing generically the GCD of two polynomials, and the resultant is the 0-th subresultant polynomial.
  The -th subresultant polynomial  of two polynomials  and  is a polynomial of degree at most  whose coefficients are polynomial functions of the coefficients of  and , and the -th principal subresultant coefficient  is the coefficient of degree  of . They have the property that the GCD of  and  has a degree  if and only if
  In this case,  is a GCD of  and  and
  Every coefficient of the subresultant polynomials is defined as the determinant of a submatrix of the Sylvester matrix of  and . This implies that subresultants "specialize" well. More precisely, subresultants are defined for polynomials over any commutative ring R, and have the following property.
  Let  be a ring homomorphism of  into another commutative ring . It extends to another homomorphism, denoted also  between the polynomials rings over  and . Then, if  and  are univariate polynomials with coefficients in  such that