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)