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resultant

compute the resultant of two polynomials

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

resultant(a, b, x)

Parameters

a, b

-

polynomials in x

x

-

name

Description

• 

The function resultant computes the resultant of the two polynomials a and b with respect to the indeterminate x.

• 

If a and b are polynomials over an integral domain, where

a=ani=1nxαi

  

and

  

 

b=bmi=1mxβi

  

then the resultant of the two polynomials a and b with respect to x is defined to be the product

  

 

anmbmni=1nj=1mαiβj

• 

The resultant can be computed from the Euclidean algorithm, or computed as the determinant of Sylvester's matrix or Bezout's matrix. For univariate and bivariate resultants over the rationals, modular methods are used for polynomials of high degree and the subresultant algorithm is used for polynomials of low degree. Otherwise Bezout's determinant is computed using minor expansion.

• 

For univariate or bivariate problems containing integer entries, the modular method produces a probabilistic result. Information on controlling the probabilistic behavior can be found in EnvProbabilistic.

• 

For efficient computation, resultant takes advantage of any factorization of a and b that is present, although no explicit factorization is attempted.

Examples

resultantax+b,cx+d,x

adbc

(1)

resultantx+a5,x+b5,x

a+b25

(2)

References

  

Buchberger, B.; Collins, G. E.; and Loos, R., eds. Computer Algebra: Symbolic & Algebraic Computation. New York: Springer-Verlag, 1982.

See Also

Algebraic[Resultant]

discrim

gcd

LinearAlgebra[BezoutMatrix]

LinearAlgebra[SylvesterMatrix]

Resultant

 


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