Definitions

Resultant

Resultant

[ri-zuhl-tnt]

In mathematics, the resultant of two monic polynomials P and Q over a field k is defined as the product

mathrm{res}(P,Q) = prod_{(x,y):,P(x)=0,, Q(y)=0} (x-y),,

of the differences of their roots, where x and y take on values in the algebraic closure of k. For non-monic polynomials with leading coefficients p and q, respectively, the above product is multiplied by

p^{deg Q} q^{deg P}.,

Computation

mathrm{res}(P,Q) = prod_{P(x)=0} Q(x),
and this expression remains unchanged if Q is reduced modulo P. Note that, when non-monic, this includes the factor q^{deg P} but still needs the factor p^{deg Q}.

  • Let P' = P mod Q. The above idea can be continued by swapping the roles of P' and Q. However, P' has a set of roots different from that of P. This can be resolved by writing prod_{Q(y)=0} P'(y), as a determinant again, where P' has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient q of Q appears.

mathrm{res}(P,Q) = q^{deg P - deg P'} cdot mathrm{res}(P',Q)
Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

  • mathrm{res}(P,Q) = (-1)^{deg P cdot deg Q} cdot mathrm{res}(Q,P)
  • mathrm{res}(Pcdot R,Q) = mathrm{res}(P,Q) cdot mathrm{res}(R,Q)
  • If P' = P + R*Q and deg P' = deg P, then mathrm{res}(P,Q) = mathrm{res}(P',Q)
  • If X, Y, P, Q have the same degree and X = a_{00}cdot P + a_{01}cdot Q, Y = a_{10}cdot P + a_{11}cdot Q,

then mathrm{res}(X,Y) = det{begin{pmatrix} a_{00} & a_{01} a_{10} & a_{11} end{pmatrix}}^{deg P} cdot mathrm{res}(P,Q)

  • mathrm{res}(P_-,Q) = mathrm{res}(Q_-,P) where P_-(z) = P(-z)

Applications

  • The resultant of a polynomial and its derivative is related to the discriminant.
  • Resultants can be used in algebraic geometry to determine intersections. For example, let

f(x,y)=0
and
g(x,y)=0
define algebraic curves in mathbb{A}^2_k. If f and g are viewed as polynomials in x with coefficients in k(y), then the resultant of f and g gives a polynomial in y whose roots are the y-coordinates of the intersection of the curves.

See also

Elimination theory

References

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