What is "discriminant"

Discriminant \Dis*crim"i*nant\, n. [L. discriminans, p. pr. of discriminare.] (Math.) The eliminant of the n partial differentials of any homogenous function of n variables. See Eliminant.

a. Serving to discriminate. n. 1 (context algebra English) An expression that gives information about the roots of a polynomial; for example, the expression ''D = b² - 4ac'' determines whether the roots of the quadratic equation ''ax² + bx + c = 0'' are real and distinct (''D'' > 0), real and equal (''D'' = 0) or complex (''D'' < 0). 2 (cx geometry English) The invariant (on the vector space of forms of degree ''d'' in ''n'' variables) that vanishes exactly when the corresponding hypersurface in P^''n''-1 is singular.

In algebra, the discriminant of a polynomial is a function of its coefficients, typically denoted by a capital 'D' or the capital Greek letter Delta (Δ). It gives information about the nature of its roots. The discriminant is zero if and only if the polynomial has a multiple root. For example, the discriminant of the quadratic polynomial

ax + bx + c 

is

Δ =  b − 4ac.

Here for real a, b and c, if Δ > 0, the polynomial has two real roots, if Δ = 0, the polynomial has one real double root, and if Δ < 0, the two roots of the polynomial are complex conjugates.

The discriminant of the cubic polynomial

ax + bx + cx + d 

is

Δ =  bc − 4ac − 4bd − 27ad + 18abcd.

For higher degrees, the discriminant is always a polynomial function of the coefficients. It becomes significantly longer for the higher degrees. The discriminant of a general quartic has 16 terms, that of a quintic has 59 terms, that of a sextic has 246 terms, and the number of terms increases exponentially with the degree. This is OEIS sequence .

If the lead coefficient a equals 1, the discriminant equals the product of the squared differences of all pairs of roots of the polynomial. Thus a polynomial has a multiple root (i.e. a root with multiplicity greater than one) in the complex numbers if and only if its discriminant is zero.

The concept also applies if the polynomial has coefficients in a field which is not contained in the complex numbers. In this case, the discriminant vanishes if and only if the polynomial has a multiple root in any algebraically closed field containing the coefficients.

As the discriminant is a polynomial function of the coefficients, it is defined as long as the coefficients belong to an integral domain R and, in this case, the discriminant is in R. In particular, the discriminant of a polynomial with integer coefficients is always an integer. This property is widely used in number theory.

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