Discriminant |
In mathematics, a polynomial P ( T ) has a discriminant, which is a polynomial function of its Coefficients, and discriminates the case of a multiple root (for which the graph of P ( x ) would touch the x -axis). This generalises to polynomials of any degree the case of a quadratic polynomial ax 2 + bx + c , for which the discriminant is b 2 − 4 ac , the quantity under the square root sign in the quadratic formula (or quadratic formula). 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.
=Discriminant of a polynomial=
The discriminant of a polynomial is a number which 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 ax 2 + bx + c is b 2 − 4 ac .
For the general definition, suppose
: p ( x ) = x n + a n −1 x n −1 + ... + a 1 x + a 0
is a polynomial with real number coefficients. The discriminant of this polynomial is defined as the Determinant of the (2 n − 1)×(2 n − 1) matrix
1 a n −1 a n −2 . . . a 0 0 . . . 0 0 1 a n −1 a n −2 . . . a 0 0 . . 0 0 0 1 a n −1 a n −2 . . . a 0 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 1 a n −1 a n −2 . . . a 0 n ( n −1) a n −1 ( n -2) a n −2 . . 1 a 1 0 0 . . . 0 0 n ( n −1) a n −1 ( n −2) a n −2 . . 1 a 1 0 0 . . 0 0 0 n ( n −1) a n −1 ( n −2) a n −2 . . 1 a 1 0 0 . 0 . . . . . . . . . . . . . . 0 0 0 0 0 n ( n −1) a n −1 a n −2 . . 1 a 1 0 0 0 0 0 0 0 n ( n −1) a n −1 a n −2 . . 1 a 1
In the case n = 4, this discriminant looks like this:
egin{vmatrix} & 1 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \ & 0 & 1 & a_3 & a_2 & a_1 & a_0 & 0 \ & 0 & 0 & 1 & a_3 & a_2 & a_1 & a_0 \ & 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \ & 0 & 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \ & 0 & 0 & 4 & 3a_3 & 2a_2 & 1a_1& 0 \ & 0 & 0 & 0 & 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 ).
One can show that, up to sign, the discriminant is equal to
:Π i < j ( r i − r j )2
where r 1, ..., r n are the (complex number) numbers such that
: p ( x ) = ( x − r 1) ( x − r 2) ... ( x − r n )
Therefore, 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 coefficient, but instead one evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree is
: a 12 a 22 − 4 a 0 a 23 − 4 a 13 + 18 a 0 a 1 a 2 − 27 a 02.
The discriminant can be defined for polynomials over arbitrary field (mathematics)s, 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.
=Discriminant of a conic=
For conic section defined by real polynomials of the form
: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0,
the discriminant is equal to
: B 2 − 4 AC ,
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).
=Case of a quadratic form=
There is a substantive generalisation, to quadratic forms Q over any field (mathematics) K of characteristic ≠ 2. These can be written as a sum of terms
: a i L i 2
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 / K 2, 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 .
See also: Discriminant of an algebraic number field|
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