Resultant - Applications

Applications

• If x and y are algebraic numbers such that (with degree of Q = n), we see that is a root of the resultant (in x) of and and that is a root of the resultant of and ; combined with the fact that is a root of, this shows that the set of algebraic numbers is a field.

• The discriminant of a polynomial is the quotient by its leading coefficient of the resultant of the polynomial and its derivative.

• Resultants can be used in algebraic geometry to determine intersections. For example, let

and

define algebraic curves in . If and are viewed as polynomials in with coefficients in, then the resultant of and is a polynomial in whose roots are the -coordinates of the intersection of the curves and of the common asymptotes parallel to the axis.

• In computer algebra, the resultant is a tool that can be used to analyze modular images of the greatest common divisor of integer polynomials where the coefficients are taken modulo some prime number . The resultant of two polynomials is frequently computed in the Lazard–Rioboo–Trager method of finding the integral of a ratio of polynomials.

• In wavelet theory, the resultant is closely related to the determinant of the transfer matrix of a refinable function.