Several versions of the resultant of two polynomials
Grace Orzech · Linear and Multilinear Algebra · 1984
Let F be any field and let f and g be in F[X]. Assume deg(g) < = n and define λ f (g) = g(C) f where C f is the companion matrix of a n −1 f(an = the leading coefficient of f). We show how λ f (g) is related to various matrices that are used to produce the resultant of f and g Connection with methods due to Bezout. Cayley, Sylvester and H. Laurent are discussed. Furthermore suppose f(X) = an (X − α1 p1 … (X − α k pk with an in F and α1, …, α k distinct elements of the algebraic closure of F For each β in (g(αi)|i = 1, …, k} let Iβ = {g(α i ) = β and write The Jordan canonical form of λ f (g) is described in terms of the p i 's and q i 's.