Diskriminimi

Diskriminimi, in Finnish often connected with the term diskriminantti or simply diskriminimi, is a mathematical quantity computed from the coefficients of a polynomial or quadratic form. It encodes information about the roots of the polynomial or the geometric type of the conic section, and it is invariant under certain changes of variables. The term in English is discriminant.

In the case of a quadratic equation ax^2 + bx + c = 0 with a ≠ 0, the discriminim is Δ = b^2 − 4ac. Its sign determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives a repeated real root, and Δ < 0 yields two complex roots (conjugates). The discriminim thus directly informs solution strategies and root multiplicities.

For higher-degree polynomials P(x) = a_n x^n + … + a_0, the discriminim generalizes so that Δ(P) = 0 if and only if P has a multiple root in its algebraic closure. For example, the cubic discriminim Δ = 18abcd − 4b^3d + b^2c^2 − 4ac^3 − 27a^2d^2, where P(x) = ax^3 + bx^2 + cx + d. In general, discriminants can be expressed in terms of resultants with the derivative P′ and become progressively more complex for higher degrees.

In the context of conic sections Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, the quantity B^2 − 4AC is a discriminim that classifies the conic: negative values indicate an ellipse, zero indicates a parabola, and positive values indicate a hyperbola (assuming nondegenerate cases). The discriminim concept connects algebra, geometry, and arithmetic, and appears in various forms across mathematics, including number theory and algebraic geometry.