Discriminant | Research & Encyclopedia Articles

Discriminant

A discriminant is a number, usually invariant under suitable transformations, which characterizes some properties of the roots of a certain quantity.

For instance, the most common application of the concept of discriminant is in the second-degree equation a x² + b x+ c=0.

In this context, the discriminant is the quantity D= b²- 4a c. According to the sign of D, such an equation can have either two distinct roots (if D is greater than zero), or only one root (in the case D=0), or no root (if D is less than zero).

The concept of discriminant is also used for polynomials. In the case of a polynomial with real coefficients, the discriminant is defined as the product of the squares of the differences of the complex roots. Since the non-real roots of such a polynomial are in conjugated pairs, the discriminant is a real number. Moreover, the definition of discriminant in the case of the second degree equation (as given above) coincides with the one of the second degree polynomial up to a multiplicative positive quantity, which does not change the sign of the discriminant.

It is possible to define a discriminant also for conic sections. In this context, the sign of the discriminant depends on whether the conic is a parabola, a hyperbola, or an ellipse.

Given a general quadratic curve A x² + B xy+ C y² + D x+ + E y+ F=0. the quantity D is known as the discriminant, where D= B²- 4 AC. The discriminant defined in this way is invariant under rotation and translation.

This invariant quantity provides a useful shortcut to determining the shape represented by a quadratic curve. If D is negative, the equation represents an ellipse, a circle (degenerate ellipse), a point (degenerate circle), or has no graph.

If D is positive, the equation represents either a hyperbola or a pair of intersecting lines (degenerate hyperbola).

If D=0, the equation represents a parabola, a line (degenerate parabola), a pair of parallel lines (degenerate parabola), or has no graph.

A very useful application of the concept of discriminant is the so-called "second derivative test." Suppose that f is a twice differentiable function of two variables and that a is a stationary point for f. In this case, the discriminant D of f at the point a is defined as the determinant of the matrix of the second derivatives evaluated at the point a (this matrix is sometimes called a Hessian matrix and its determinant is called Hessian determinant).

If D is greater than zero, f_xx at a is greater than zero, and f_xx+ f_yy at a is greater than zero, then the point a is a relative minimum.

If D is greater than zero, f_xx at a is less than zero, and f_xx+ f_yy at a is less than zero, then the point a is a relative maximum.

If D is less than zero, the point a is a saddle point.

If D=0, higher order tests must be used to determine the local behavior of the function f.

The concept of discriminant can also be used for: binary quadratic forms, elliptic curves, metrics, modules, quadratic fields, and quadratic forms. The interested reader may look at the following books: H. Cohn, Advanced Number Theory (New York, Dover, 1980); D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (New York, Chelsea, 1999); J. Silverman, The Arithmetic of Elliptic Curves (New York, Springer-Verlag, 1986).