Discriminant

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them.

The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.

In terms of the roots, the discriminant is equal to It is thus the square of the Vandermonde polynomial times

It makes clear that if the polynomial has a multiple root, then its discriminant is zero, and that, in the case of real coefficients, if all the roots are real and simple, then the discriminant is positive.

The discriminant of a polynomial is, up to a scaling, invariant under any projective transformation of the variable.

When one is only interested in knowing whether a discriminant is zero (as is generally the case in algebraic geometry), these properties may be summarised as: This is often interpreted as saying that

This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.

For higher degrees, there may be monomials which satisfy above rules and do not appear in the discriminant.

are permitted to be zero, the polynomials A(x, 1) and A(1, y) may have a degree smaller than n. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree n. This means that the discriminants must be computed with

Let V be such a curve or hypersurface; V is defined as the zero set of a multivariate polynomial.

Viewing f as a univariate polynomial in Y with coefficients depending on X, then the discriminant is a polynomial in X whose roots are the X-coordinates of the singular points, of the points with a tangent parallel to the Y-axis and of some of the asymptotes parallel to the Y-axis.

The polynomial A defines a projective hypersurface, which has singular points if and only the n partial derivatives of A have a nontrivial common zero.

In the case of a homogeneous bivariate polynomial of degree d, this general discriminant is

Several other classical types of discriminants, that are instances of the general definition are described in next sections.

The multivariate resultant of the partial derivatives of Q is equal to its Hessian determinant.

In other words, the discriminant of a quadratic form over a field K is an element of K/(K×)2, the quotient of the multiplicative monoid of K by the subgroup of the nonzero squares (that is, two elements of K are in the same equivalence class if one is the product of the other by a nonzero square).

Over the rational numbers, a discriminant is equivalent to a unique square-free integer.

The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field).

In this case, either the surface may be decomposed in planes, or it has a unique singular point, and is a cone or a cylinder.

A conic section is a plane curve defined by an implicit equation of the form where a, b, c, d, e, f are real numbers.

If the discriminant is positive, the curve is a hyperbola, or, if degenerated, a pair of intersecting lines.

be a polynomial of degree two in three variables that defines a real quadric surface.

depends on three variables, and consists of the terms of degree two of P; that is Let us denote its discriminant by

and the surface has real points, it is either a hyperbolic paraboloid or a one-sheet hyperboloid.

In both cases, this is a ruled surface that has a negative Gaussian curvature at every point.

More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.

The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L.

Not every integer can arise as a discriminant of an integral binary quadratic form.