Discriminanti

Discriminanti (plural) is a term used in mathematics to refer quantities associated with polynomials or equations that signal degeneracies or special configurations of roots or singularities. In univariate algebra, a discriminante measures how roots of a polynomial relate to each other and whether multiple roots occur.

For a polynomial f(x) = a_n x^n + … + a_0 with leading coefficient a_n, the discriminante Δ(f) is defined as Δ(f) = a_n^{2n-2} ∏_{i<j} (r_i − r_j)^2, where r_i are the roots of f in its splitting field. A key property is that Δ(f) = 0 if and only if f has a repeated root. In the familiar quadratic case f(x) = ax^2 + bx + c, the discriminante is Δ = b^2 − 4ac, which determines the real or complex nature of the roots and whether they are distinct.

Discriminants generalize beyond single-variable polynomials. They define the discriminant locus in families of curves: the set of parameter values for which the corresponding curve has a singular point. In algebraic geometry, this helps identify degenerations of families of shapes. In number theory, the discriminant D_K of a number field K encodes ramification of primes in the field and relates to the ring of integers, capturing arithmetic complexity.

Overall, discriminanti provide a compact way to summarize when algebraic objects lose generic behavior—such as acquiring repeated roots, developing singularities, or changing arithmetic properties—across different areas of mathematics.