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) =
A multivariate generalization exists (the Macaulay resultant) for systems of polynomials in several variables. Such resultants