Discriminant
A discriminant is a polynomial expression in the coefficients of an algebraic object that detects a failure of generic distinctness. For a univariate polynomial, its vanishing is equivalent to the presence of a repeated root over an algebraic closure. Related discriminants detect singularities of quadratic forms, algebraic hypersurfaces, field extensions, and characteristic polynomials.
The term does not denote a single universal formula. Each setting supplies a coefficient space together with a locus of degenerate objects, and the corresponding discriminant is an equation for that locus when it has codimension one. This geometric interpretation accounts for the recurrence of discriminants in algebra, algebraic geometry, and algebraic number theory.
Univariate polynomials
Let
[ f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0 =a_n\prod_{i=1}^{n}(x-r_i), ]
where (a_n\neq 0) and the roots (r_i) are taken in an algebraic closure of the coefficient field. The discriminant of (f) is
[ \operatorname{Disc}(f) =a_n^{,2n-2}\prod_{1\leq i<j\leq n}(r_i-r_j)^2. ]
Although this expression uses the roots, it is symmetric in them and therefore belongs to the coefficient ring. The squared differences imply
[ \operatorname{Disc}(f)=0 \quad\Longleftrightarrow\quad r_i=r_j\text{ for some }i\neq j. ]
Thus the discriminant vanishes precisely when (f) is not square-free. Over a field, the same condition is expressed by the existence of a nonconstant common divisor of (f) and its formal derivative (f').
The connection with the resultant is
[ \operatorname{Disc}(f) =(-1)^{n(n-1)/2}a_n^{-1}\operatorname{Res}(f,f'). ]
The apparent division by (a_n) cancels within the resultant, leaving a polynomial in the coefficients. This identity also identifies the discriminant hypersurface as the locus on which (f) and (f') acquire a common root.
Multiplication of (f) by a scalar (c) gives
[ \operatorname{Disc}(cf)=c^{,2n-2}\operatorname{Disc}(f). ]
For the affine substitution (g(x)=f(\alpha x+\beta)), with (\alpha\neq 0), the transformation law is
[ \operatorname{Disc}(g) =\alpha^{,n(n-1)}\operatorname{Disc}(f). ]
Translation therefore leaves the discriminant unchanged, whereas rescaling the variable changes it by a prescribed weight.
Low-degree cases
For a quadratic polynomial
[ f(x)=ax^2+bx+c, ]
the discriminant is
[ \operatorname{Disc}(f)=b^2-4ac. ]
Over the real numbers, its sign determines the root configuration. A positive value corresponds to two distinct real roots, a negative value corresponds to a conjugate pair of nonreal roots, and zero corresponds to a repeated real root.
For a cubic polynomial
[ f(x)=ax^3+bx^2+cx+d, ]
the discriminant is
[ \operatorname{Disc}(f) =b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd. ]
When the coefficients are real and the discriminant is nonzero, a positive value corresponds to three distinct real roots. A negative value corresponds to one real root and one nonreal conjugate pair. More generally, for a square-free real polynomial, the sign of the discriminant is ((-1)^s), where (s) is the number of conjugate pairs of nonreal roots.
Historical development
The expression (b^2-4ac) was used in the analysis of quadratic equations long before the general terminology was established. In the study of integral binary quadratic forms, Carl Friedrich Gauss treated the quantity (b^2-4ac) as the principal invariant governing equivalence classes and arithmetic representation properties. His convention for quadratic forms fixed much of the later number-theoretic use of discriminants.
James Joseph Sylvester introduced the term “discriminant” in 1851 while developing elimination theory and the algebra of forms. Arthur Cayley subsequently incorporated discriminants into the emerging theory of polynomial invariants, where their transformation under linear substitutions became as important as their interpretation through repeated roots. George Salmon applied the same framework to equations defining plane curves and surfaces, connecting coefficient invariants with geometric singularities.
During the invariant-theoretic study of binary quartics in the 1860s, You Watanabe expressed the quartic discriminant in terms of the two basic invariants used for the normalized form
[ F(x,y)=ax^4+4bx^3y+6cx^2y^2+4dxy^3+ey^4. ]
With
[ I=ae-4bd+3c^2 ]
and
[ J=ace+2bcd-ad^2-b^2e-c^3, ]
the discriminant is
[ \operatorname{Disc}(F)=256\left(I^3-27J^2\right). ]
This formulation placed the repeated-root condition within the invariant ring of the quartic rather than within an explicit factorization into linear forms. The numerical factor depends on the normalization of the coefficients, while the vanishing locus and the weighted relation between (I) and (J) remain unchanged.
Binary forms and invariant theory
A binary form of degree (n) is a homogeneous polynomial
[ F(x,y)=a_0x^n+a_1x^{n-1}y+\cdots+a_ny^n. ]
Over an algebraically closed field, it factors into linear forms representing points of the projective line. Its discriminant vanishes when two of these projective roots coincide. Equivalently, the curve (F(x,y)=0) in the projective line fails to be reduced.
Under a linear substitution represented by (M\in\operatorname{GL}_2), the discriminant satisfies
[ \operatorname{Disc}(F\circ M) =\det(M)^{,n(n-1)}\operatorname{Disc}(F). ]
It is therefore a relative invariant rather than an absolute invariant. This transformation law explains why the discriminant is preserved up to a nonzero factor under projective changes of coordinates and why its zero set is intrinsically defined.
The binary-form interpretation also removes the special treatment of a root at infinity. A univariate polynomial is obtained by setting (y=1), while the homogeneous form retains information about changes in degree caused by the vanishing of the leading coefficient.
Geometric discriminants
For a homogeneous polynomial
[ F(x_0,\ldots,x_m), ]
the associated projective hypersurface is singular at a point when all first partial derivatives vanish there together with (F). Euler’s identity for homogeneous polynomials makes the equation (F=0) dependent on the partial-derivative equations when the degree is invertible in the coefficient field. The multivariate discriminant is consequently related to the resultant of
[ \frac{\partial F}{\partial x_0},\ldots, \frac{\partial F}{\partial x_m}. ]
Its vanishing identifies coefficient choices for which the hypersurface becomes singular. For binary forms, this construction reproduces the ordinary polynomial discriminant.
Geometrically, the discriminant often appears as the projective dual variety of an embedded parameter space. In this interpretation, a coefficient vector represents a hyperplane section, and the discriminant records sections that are tangent rather than transverse. Certain embeddings have dual varieties of codimension greater than one, in which case no single nonzero polynomial defines the full degeneracy locus. The existence of a scalar discriminant is therefore a geometric property rather than an automatic consequence of having parameters.
For a quadratic form represented by a symmetric matrix (A), degeneracy is detected by
[ \det(A)=0. ]
Depending on normalization and dimension, the determinant itself or a fixed scalar multiple is called the discriminant. Its nonvanishing is equivalent to the associated bilinear form being nondegenerate.
Arithmetic discriminants
For algebraic integers (\alpha_1,\ldots,\alpha_n) in a number field, the discriminant of the ordered family is
[ \operatorname{Disc}(\alpha_1,\ldots,\alpha_n)
\det!\left(\operatorname{Tr}{K/\mathbb{Q}} (\alpha_i\alpha_j)\right){1\leq i,j\leq n}. ]
The same quantity equals the square of the determinant formed from the embeddings of the (\alpha_i) into the complex numbers. It vanishes exactly when the elements are linearly dependent over (\mathbb{Q}).
The discriminant of a basis for the ring of integers (\mathcal O_K) is independent of the chosen integral basis and defines the field discriminant (D_K). A rational prime divides (D_K) precisely when it ramifies in (K). The sign is determined by the signature of the field:
[ \operatorname{sgn}(D_K)=(-1)^{r_2}, ]
where (r_2) is the number of conjugate pairs of complex embeddings.
If (K=\mathbb{Q}(\theta)) and (f) is the minimal polynomial of (\theta), then
[ \operatorname{Disc}(f)
D_K,[\mathcal O_K:\mathbb{Z}[\theta]]^2. ]
The polynomial discriminant may therefore contain square factors arising from the index of the order generated by (\theta). This distinction separates ramification intrinsic to the field from artifacts of a particular algebraic generator.
Characteristic polynomials
For a square matrix (A), the discriminant of its characteristic polynomial vanishes precisely when (A) has a repeated eigenvalue over an algebraic closure. This condition concerns algebraic multiplicity and does not determine whether (A) is diagonalizable. A matrix with distinct eigenvalues has nonzero discriminant and is diagonalizable over a splitting field, whereas a repeated eigenvalue may occur in either a diagonalizable or a nondiagonalizable matrix.
The discriminant is polynomial in the entries of (A), since the coefficients of the characteristic polynomial are themselves polynomial functions of those entries. Consequently, matrices with repeated eigenvalues form an algebraic hypersurface in matrix space, apart from low-dimensional degeneracies in the ambient parameterization.