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Discriminant

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Discriminant[poly, var]
    gives the discriminant of poly with respect to var: up to sign and
    leading-coefficient scaling, Resultant[poly, D[poly, var], var] /
    lc[poly, var].  Vanishes iff poly has a repeated root in var.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Discriminant[a x^2 + b x + c, x]
Out[1]= b^2 - 4 a c

In[2]:= Discriminant[5 x^4 - 3 x + 9, x]
Out[2]= 23273325

In[3]:= Discriminant[(x-1)(x-2)(x-3), x]
Out[3]= 4

In[4]:= Discriminant[(x-1)(x-2)(x-1), x]
Out[4]= 0

Implementation notes

Algorithm. builtin_discriminant (in src/poly/poly.c) computes Disc_x(p) from the standard resultant identity. After confirming p is a polynomial in x (PolynomialQ) and expanding it, it reads the degree n (returning 0 for degree 0/1). It then forms the derivative p' with poly_derivative, computes R = Res_x(p, p') via resultant_internal, and applies the closed form

Disc = (-1)^{n(n-1)/2} · Res(p, p') / a_n,

where a_n is the leading coefficient (obtained with get_coeff). The quotient is built as Times[sign·R, a_n^{-1}] and the final result is expr_expanded.

Data structures. Plain Expr* polynomials throughout; the heavy lifting is the subresultant/Euclidean resultant_internal (see Resultant). Degree and coefficient queries use get_degree_poly/get_coeff.

Complexity / limits. Dominated by the resultant computation, which is polynomial in n and the coefficient sizes but can be expensive for large multivariate inputs.

  • Protected, Listable.
  • Computes the discriminant of polynomial poly with respect to var.
  • The discriminant is zero if and only if the polynomial has multiple roots.
  • Derived symbolically utilizing the formula $D = \frac{(-1)^{n(n-1)/2}}{a_n} Resultant(P, P', var)$.

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Discriminant[x^2 - 5 x + 6, x]
Out[1]= 1

In[1]:= Discriminant[a x^2 + b x + c, x]
Out[1]= b^2 - 4 a c

In[1]:= Discriminant[x^3 + p x + q, x]
Out[1]= -4 p^3 - 27 q^2

In[1]:= Discriminant[x^4 + 1, x]
Out[1]= 256

Notes

Discriminant[poly, var] is, up to sign and leading-coefficient scaling, Resultant[poly, D[poly, var], var] / lc[poly, var], and it vanishes exactly when poly has a repeated root in var. The first example has distinct roots 2 and 3, so its discriminant is a nonzero constant. The second recovers the familiar b^2 - 4 a c from the quadratic formula. The third gives the classical depressed-cubic discriminant -4 p^3 - 27 q^2, whose sign distinguishes three real roots from one real and two complex roots. The fourth shows x^4 + 1, which has four distinct complex roots and discriminant 256.