What are the algorithms to compute the resultant ${\rm Res}(f(x),g(x),x)$ of two univariate polynomials $f,g$?

Question

I am trying to implement an algorithm for computing Res(f(x),g(x),x) where f(x) and g(x) uni variate polynomials with integer coefficients. Could any one list the various algorithms for computing Res(f(x),g(x),x) along with a brief comparison (e.g. time complexity analysis)? I know that the resultant is the determinant of the Sylvester matrix. But is this the best way for computing it?

Thanks in advance

Answer 1

The resultant is usually computed by a variant of Euclid's algorithm, e.g.: http://www.csd.uwo.ca/~eschost/publications/CS829-lecture3.pdf

For Z[x], a modular method can be used to reduce blowup, and there are also fast algorithms for large problems. See "Algorithms for Computer Algebra" by Geddes et al for a nice introduction or "Modern Computer Algebra" by von zur Gathen and Gerhard for a comprehensive reference.