Diszkrimináns

Diszkrimináns is a mathematical quantity associated with polynomials, quadratic forms, or algebraic expressions that provides information about the roots or geometric properties of the object. In algebra, the discriminant vanishes exactly when the polynomial has a multiple root in the complex numbers, indicating a degeneracy in its root structure. For higher degree polynomials this idea is generalized by defining the discriminant through the resultant of the polynomial and its derivative.

For a quadratic polynomial f(x) = ax^2 + bx + c with a ≠ 0, the discriminant is Δ = b^2 − 4ac. The sign of Δ determines the real root configuration: Δ > 0 yields two distinct real roots, Δ = 0 a repeated real root, and Δ < 0 two complex conjugate roots. This simple case highlights the core role of the discriminant in linking coefficients to the nature of solutions.

In general, the discriminant Δ(f) of a polynomial f is zero if and only if f has a multiple root in its algebraic closure. It can be computed as the resultant of f and its derivative f′ (up to an overall factor), and its nonvanishing implies that f has distinct roots in the complex plane. Discriminants extend to other objects, such as binary quadratic forms where Δ = B^2 − 4AC classifies conic sections up to projective transformations, and to number fields where the field discriminant encodes arithmetic properties of the ring of integers.

Overall, the discriminant is a compact invariant that signals degeneracy, root multiplicity, or qualitative changes in the associated geometric or arithmetic structure.