Discriminant

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In mathematics, a discriminant is an expression that discriminates qualities of algebraic structures. The concept applies to polynomials, conic sections, quadratic forms, and algebraic number fields.

For a polynomial P(x) = a₀ + a₁x + a₂x² + ... , the discriminant is a quantity D = D(a₀,a₁,a₂,...) that equals 0 precisely for those P(x) that have at least one multiple root.

For a quadratic equation, the discriminant is the square-rooted section of the Quadratic Formula because you can use it to discriminate between whether the given quadratic has two real solutions, one solution, or no real solutions.

For instance, the quadratic polynomial P(x) = ax² + bx + c has discriminant D = b² − 4ac, which is the quantity under the square root sign in the quadratic formula. For real numbers a, b, c, one has:

When D > 0 , P(x) has two distinct real roots $x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ , and its graph crosses the x-axis twice.
When D = 0, P(x) has two coincided real roots $x_1=x_2=-\frac{b}{2a}$ , and its graph is tangent to the x-axis.
When D < 0 , P(x) has no real roots, and its graph lies strictly above or below the x-axis.

Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.

1 Discriminant of a polynomial
2 Discriminant of a conic section
3 Discriminant of a quadratic form
4 Discriminant of an algebraic number field

[edit] Discriminant of a polynomial

A discriminant of a polynomial is a number that can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discriminant of the polynomial ax² + bx + c is b² − 4ac.

For the general definition, suppose

$p(x)=a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_1 x+a_0$

is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix

$\left(\begin{matrix} & a_n & a_{n-1} & a_{n-2} & \ldots & a_0 & 0 & \ldots & \ldots & 0 \\ & 0 & a_n & a_{n-1} & a_{n-2} & \ldots & a_0 & 0 & \ldots & 0 \\ & \vdots\ &&&&&&&&\vdots\\ & 0 & 0& \ldots\ & 0 & a_n & a_{n-1} & a_{n-2} & \ldots & a_0 \\ & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \ldots\ & 1a_1 & 0 & \ldots &\ldots & 0 \\ & 0 & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \ldots\ & 1a_1 & 0 & \ldots & 0 \\ & \vdots\ &&&&&&&&\vdots\\ & 0 & 0 & \ldots & 0 & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \ldots\ & 1a_1 \\ \end{matrix}\right)$

In the case n = 4, this discriminant looks like this:

$\begin{vmatrix} & a_4 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \\ & 0 & a_4 & a_3 & a_2 & a_1 & a_0 & 0 \\ & 0 & 0 & a_4 & a_3 & a_2 & a_1 & a_0 \\ & 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \\ & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \\ & 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1& 0 \\ & 0 & 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 \\ \end{vmatrix}$

The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x), where $p'(x)$ is the derivative of p(x).

One can show that, up to sign, the discriminant is equal to

$a_n^{2n-2}\prod_{i<j}{(r_i-r_j)^2}$

where r₁, ..., r_n are the complex roots (counting multiplicity) of the polynomial p(x):

$\begin{matrix}p(x)&=&a_n x^n+a_{n-1}x^{n-1}+\ldots+a_1 x+a_0\\ &=&a_n(x-r_1)(x-r_2)\ldots (x-r_n)\end{matrix}$

In fact, some authors define the discriminant by that formula, then show that the sign difference to the resultant is (−1)^{n(n −1)/2} .

It is clear from this second definition that, p has a multiple root if and only if the discriminant is zero. Note, however, that this multiple root can be complex.

In order to compute discriminants, one does not evaluate the above determinant each time for different coefficients, but instead evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree $p (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0$ is

$a_1^2 a_2^2-4a_0 a_2^3-4a_1^3 a_3+18a_0 a_1 a_2 a_3-27a_0^2 a_3^2$

The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots r_i remains valid; the roots have to be taken in some splitting field of the polynomial.

[edit] Discriminant of a conic section

For a conic section defined by the real polynomial:

ax² + bxy + cy² + dx + ey + f= 0,

the discriminant is equal to

b² − 4ac,

and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will not work for degenerate cases (when the polynomial factorises).

[edit] Discriminant of a quadratic form

There is a substantive generalisation to quadratic forms Q over any field K of characteristic ≠ 2. These can be written as a sum of terms

a_iL_i²

where the L_i are linear forms and 1 ≤ i ≤ n where n is the number of variables. Then the discriminant is the product of the a_i, taken in K/K², and is then well defined (i.e., up to squares). A more invariant way to say this is as (the class of) the determinant of a symmetric matrix for Q.