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. The polynomial x² + bx + c for instance has the discriminant b² - 4c.

For the general definition, suppose

p(x) = xⁿ + a_n-1x^n-1 + ... + a₁x + a₀

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

  1     an-1     an-2      .         .        .    a0       0        .   .   .   0
  0     1        an-1     an-2       .        .    .       a0        0   .   .   0
  0     0        1        an-1     an-2       .    .       .        a0   0   .   0
  .     .        .        .         .        .    .
  .     .        .        .         .        .    .
  0     0        0        0         0        1    an-1    an-2       .   .   .  a0
  n  (n-1)an-1 (n-2)an-2   .         .       1a1   0        0        .   .   .   0
  0     n      (n-1)an-1 (n-2)an-2   .        .   1a1       0        0   .   .   0
  0     0        n       (n-1)an-1 (n-2)an-2  .    .       1a1       0   0   .   0
  .     .        .        .         .        .    .
  .     .        .        .         .        .    .
  0     0        0        0         0        n  (n-1)an-1  an-2      .   .  1a1  0
  0     0        0        0         0        0    n      (n-1)an-1  an-2 .   .  1a1

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

   1    a3   a2   a1   a0   0    0
   0    1    a3   a2   a1   a0   0 
   0    0    1    a3   a2   a1   a0
   4    3a3  2a2  1a1   0   0    0
   0    4    3a3  2a2  1a1  0    0   
   0    0    4    3a3  2a2  1a1  0
   0    0    0    4    3a3  2a2  1a1

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 r₁, ..., r_n are the (complex) numbers such that

p(x) = (x - r₁) (x - r₂) ... (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 a₁²a₂² - 4a₀a₂³ -4a₁³ + 18 a₀a₁a₂ - 27a₀².

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.