discriminants

Discriminants are algebraic expressions associated with polynomials and quadratic forms that encode information about their roots and their geometry. For a univariate polynomial f(x) of degree n with leading coefficient a_n and roots r_1, ..., r_n, the discriminant Disc(f) is a_n^(2n-2) times the product over all pairs i<j of (r_i - r_j)^2. The discriminant vanishes exactly when f has a multiple root over the complex numbers, and hence when roots coincide. It remains unchanged under permutations of the roots and, under an affine change of variable x -> alpha x + beta, scales by alpha^(n(n-1)).

In the special case of a quadratic f(x) = a x^2 + b x + c, the discriminant is Delta = b^2 - 4 a c. Its value determines the real root structure: two distinct real roots if Delta > 0, a repeated real root if Delta = 0, and two complex conjugate roots if Delta < 0.

For a cubic f(x) = a x^3 + b x^2 + c x + d, the discriminant is Delta = 18 a b c d - 4 b^3 d + b^2 c^2 - 4 a c^3 - 27 a^2 d^2; it vanishes precisely when f has a multiple root. Higher-degree discriminants are more intricate, but they share the same core property: Delta = 0 signals a degeneracy in the root structure.

Discriminants also arise for quadratic forms and plane conics. A binary quadratic form A x^2 + B x y + C y^2 has discriminant Delta = B^2 - 4 A C; for a conic A x^2 + B x y + C y^2 + D x + E y + F = 0, the discriminant B^2 - 4 A C helps classify the conic and detect degeneracies. In broader contexts, discriminants reflect singularities in algebraic geometry and ramification in number theory, linking roots, geometry, and arithmetic.