Resultant - Properties

Properties

  • The resultant is zero if and only if the two polynomials have a common root in an algebraically closed field containing the coefficients.
  • Since the resultant is a polynomial with integer coefficients in term of the coefficients of P and Q, it follows that
    • The resultant is well defined for polynomials over any commutative ring.
    • If h is a homomorphism of the ring of the coefficients into another commutative ring, which preserve the degrees of P and Q, then the resultant of the image by h of P and Q is the image by h of the resultant of P and Q.
  • The resultant of two polynomials with coefficients in a integral domain is null if and only if they have a common divisor of positive degree.
  • res(P,Q) = (-1)degPdegQres(Q,P)
  • res(PR,Q)=res(P,Q)res(R,Q)
  • If P'=P+RQ and degP'=degP, then res(P,Q)=res(P',Q).
  • If X, Y, P, Q have the same degree and X=a00P+a01Q, Y=a10P+a11Q,
then
  • res(P-,Q)=res(Q-,P) where P-(z)=P(-z)

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