resultant

Resultant is a term used in multiple contexts, most notably in algebra and physics. In algebra, the resultant of two univariate polynomials over a field is a scalar function of their coefficients that vanishes precisely when the polynomials have a common root in some extension field. It serves as an elimination tool in solving systems of polynomial equations.

For polynomials P and Q of degrees m and n, the resultant can be defined as the determinant of the Sylvester matrix built from the coefficients of P and Q. Equivalently, it can be expressed as a product of Q evaluated at the roots of P, up to a factor involving the leading coefficients. The resultant is zero if and only if gcd(P, Q) ≠ 1, i.e., the polynomials share a common root.

In applications, resultants are used to eliminate a variable from systems of polynomial equations, producing a single-variable equation whose roots correspond to common solutions. They are central to elimination theory, computational algebra, and algebraic geometry, and are implemented in many computer algebra systems as a tool for solving polynomial systems.

A related concept is the discriminant, which for a polynomial f is (up to a power of the leading coefficient) the resultant of f and its derivative f'. The discriminant vanishes precisely when f has a repeated root, linking it to the geometry and singularities of the polynomial’s root structure.

In physics, the term resultant also appears in the context of vector addition. The resultant of a set of forces is the net force obtained by vector summation; its magnitude and direction summarize the combined effect. If the resultant is zero, the forces are in equilibrium.