Resultants

The resultant is a classical algebraic construction that produces a single scalar from two polynomials whose value encodes whether the polynomials have a common root. In the univariate case, let F be a field and f(x), g(x) ∈ F[x] with degrees m = deg f and n = deg g, and leading coefficients a_m, b_n. The resultant Res(f,g) can be defined as the determinant of the Sylvester matrix built from the coefficients of f and g, or equivalently by the product formula Res(f,g) = a_m^n ∏_{i=1}^m g(r_i) where r_i are the roots of f in an algebraic closure of F (counted with multiplicity). Equivalently, Res(f,g) = (-1)^{mn} b_n^m ∏_{j=1}^n f(s_j) where s_j are the roots of g. A fundamental property is that Res(f,g) = 0 if and only if f and g have a common root in the algebraic closure, i.e., gcd(f,g) ≠ 1.

The resultant is homogeneous in the coefficients and satisfies Res(cf, g) = c^n Res(f,g) and Res(f, cg) = c^m Res(f,g) for any c ∈ F. It also connects to the discriminant: disc(f) = (-1)^{m(m-1)/2} a_m^{-(m-1)} Res(f, f').

A multivariate generalization exists (the Macaulay resultant) for systems of polynomials in several variables. Such resultants are used in elimination theory and computational algebraic geometry to determine the existence of common solutions and to eliminate variables in solving polynomial systems. They provide a bridge between root structure and coefficient data, enabling algorithmic approaches in symbolic computation.