Resultanten

Resultanten, a term used in some languages for the mathematical notion connue as the resultant, is a polynomial invariant in the coefficients of a system of polynomials that vanishes precisely when the polynomials have a common root. It plays a central role in elimination theory and in characterizing when systems of equations in several variables admit nontrivial solutions. The classical one-variable case leads to concrete determinant formulas, while the multivariate case gives rise to more abstract constructions that define a hypersurface in the space of coefficients.

In the univariate case, let f and g be polynomials in one variable x over a field, with f of degree m and g of degree n. The resultant Res(f,g) is defined as the determinant of the Sylvester matrix of f and g, a (m+n)×(m+n) matrix built from the coefficients of f and g. Equivalently, if r1, …, rm are the roots of f in an algebraic closure and g(r) denotes the evaluation of g at r, then Res(f,g) = a_m^n ∏ g(r_i) = (−1)^{mn} b_n^m ∏ f(s_j), where a_m and b_n are the leading coefficients and s_j the roots of g. Consequently Res(f,g) = 0 if and only if f and g have a common root. The resultant is homogeneous in the coefficients and is invariant under certain linear changes of variables.

Multivariate resultants extend the idea to systems f1 = … = fk = 0 in several variables, producing a polynomial in the coefficients that vanishes exactly when the system has a nontrivial common solution in projective space. Constructions include Macaulay’s resultant and Dixon-type variants.

Applications include elimination of variables, deriving conditions for solvability, and relating to the discriminant, which is a special case Res(f, f′) for univariate polynomials.