Resultant

The resultant of the polynomials

(1)

and

(2)

of polynomial degrees and with roots and , respectively, is denoted or and is also called the eliminant. It is defined by

(3)

(Trott 2006, p. 26).

For nonzero polynomials of positive degree over a field, the resultant vanishes iff their polynomial greatest common divisor has positive degree. It is the zeroth subresultant and the zeroth principal subresultant coefficient (El Kahoui 2003, pp. 284-285 and 290-291).

The resultant is also given by the determinant of the corresponding Sylvester matrix.

Kronecker gave a series of lectures on resultants during the summer of 1885 (O'Connor and Robertson 2005).

An important application of the resultant is the elimination of one variable from a system of two polynomial equations (Trott 2006, p. 26).

The resultant of two polynomials can be computed using the Wolfram Language function Resultant[poly1, poly2, var]. This command accepts the following methods: Automatic, SylvesterMatrix, BezoutMatrix, Subresultants, and Modular, where the optimal choice depends dramatically on the concrete polynomial pair under consideration and typically requires some experimentation. For high-order univariate polynomials over the integers, the option setting Modular is frequently the fastest (Trott 2006, p. 29).

There exists an algorithm similar to the Euclidean algorithm for computing resultants (Pohst and Zassenhaus 1989).

Resultants for a few simple pairs of polynomials include

			(4)
			(5)
			(6)

Given and , then

(7)

is a polynomial of degree , having as its roots all sums of the form .