Given a polynomial

(1)

of degree  with roots , , ...,  and a polynomial

(2)

of degree  with roots , , ..., , the resultant , also denoted  and also called the eliminant, is defined by

(3)

(Trott 2006, p. 26).

Amazingly, 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 .

REFERENCES:

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Apostol, T. M. "The Resultant of the Cyclotomic Polynomials  and ." Math. Comput. 29, 1-6, 1975.

Bikker, P. and Uteshev, A. Y. "On the Bézout Construction of the Resultant." J. Symb. Comput. 28, 45-88, 1999.

Bykov, V.; Kytmanov, A.; Lazman, M.; and Passare, M. (Eds.). Elimination Methods in Polynomial Computer Algebra. Dordrecht, Netherlands: Kluwer, 1998.

Childs, L. A Concrete Introduction to Higher Algebra. New York: Springer-Verlag, 1992.

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Gelfand, I. M.; Kapranov, M.; and Zelevinsky, A. Discriminants, Resultants and Multidimensional Resultants. Boston: Birkhäuser, 1994.

Maculay, F. S. The Algebraic Theory of Modular Systems. Cambridge: Cambridge University Press, 1916.

O'Connor, J. J. and Robertson, E. F. "Henry Burchard Fine." August 2005. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fine_Henry.html.

Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Number Theory. Cambridge, England: Cambridge University Press, 1989.

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Sturmfels, B. In Applications of Computational Algebraic Geometry. American Mathematical Society Short Course January 6-7, 1997 San Diego, California (Ed. D. A. Cox and B. Sturmfels). Providence, RI: Amer. Math. Soc., 1997.

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