discriminanten

Discriminanten is a mathematical concept that attaches a single quantity to a polynomial or form, encoding information about its roots or singularities. For a univariate polynomial with coefficients in a field, the discriminant Δ(P) is a polynomial in the coefficients that vanishes precisely when the polynomial has a multiple root in its algebraic closure. Thus, a nonzero discriminant indicates that all roots are simple.

In the quadratic case, for P(x) = ax^2 + bx + c, the discriminant is Δ = b^2 − 4ac. Its sign over the real numbers determines the nature of the roots: Δ > 0 yields two distinct real roots, Δ = 0 a repeated real root, and Δ < 0 two complex conjugate roots.

For polynomials of higher degree, the discriminant can be expressed as Δ(P) = a_n^{2n−2} ∏_{i<j} (r_i − r_j)^2, where r_i are the roots and a_n the leading coefficient. Equivalently, Δ(P) = (−1)^{n(n−1)/2} Res(P, P′), where Res denotes the resultant. The discriminant therefore vanishes exactly when the polynomial and its derivative share a root, i.e., when the polynomial has a multiple root.

Beyond univariate polynomials, discriminanten are defined for forms and systems and can determine when a family of polynomials yields singular fibers. In geometry, the discriminant locus in parameter space marks coefficients for which the corresponding hypersurface is singular.

The discriminant also appears in the study of quadratic forms and conics; for a homogeneous quadratic form Ax^2 + Bxy + Cy^2, the discriminant B^2 − 4AC classifies conic sections and degeneracy. Overall, discriminanten provide a compact test for root multiplicities and singular behavior across many contexts.