diskrimineerivat

Diskrimineerivat, commonly translated as the discriminant, is a polynomial expression in the coefficients of a polynomial that encodes information about its roots. For a polynomial P(x) = a_n x^n + ... + a_0 with a_n ≠ 0, the discriminant D(P) can be defined as D(P) = a_n^{2n−2} ∏_{i<j} (α_i − α_j)^2, where α_1, ..., α_n are the roots of P (in the complex plane). Consequently, D(P) = 0 precisely when P has a multiple root. The discriminant is thus a polynomial in the coefficients of P and vanishes exactly at parameter values where the roots fail to be simple.

For a quadratic polynomial P(x) = ax^2 + bx + c with a ≠ 0, the discriminant is D = b^2 − 4ac. When the coefficients are real, D > 0 indicates two distinct real roots, D = 0 indicates a repeated real root, and D < 0 indicates two complex conjugate roots. For polynomials of higher degree, the discriminant generalizes this role: it can be computed as the resultant of P and its derivative P′ (up to a power of a_n) and serves as a criterion for multiple roots.

In broader terms, the discriminant is central in algebraic geometry and number theory. It defines the discriminant locus of a family of polynomials—values of the parameters for which the corresponding algebraic variety becomes singular. It also relates to the behavior of roots under perturbations and to the ramification properties in number fields.

See also: discriminant of a conic or polynomial, resultant, Sylvester matrix, singularities in algebraic geometry, discriminant locus.