Discriminant

Template:Short description Template:Distinguish {{#invoke:other uses|otheruses}} Template:More citations needed

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

The discriminant of the quadratic polynomial <math>ax^2+bx+c</math> is

<math>b^2-4ac,</math>

the quantity which appears under the square root in the quadratic formula. If <math>a\ne 0,</math> this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.

More generally, the discriminant of a univariate polynomial of positive degree is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non-real roots is a multiple of 4 (including none), and negative otherwise.

Several generalizations are also called discriminant: the discriminant of an algebraic number field; the discriminant of a quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially equivalent).

Origin

The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.<ref>Template:Cite journal
Sylvester coins the word "discriminant" on page 406.</ref>

Definition

Let

<math>A(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0</math>

be a polynomial of degree Template:Math (this means <math>a_n\ne 0</math>), such that the coefficients <math>a_0, \ldots, a_n</math> belong to a field, or, more generally, to a commutative ring. The resultant of Template:Math with its derivative,

<math>A'(x) = na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots+a_1,</math>

is a polynomial in <math>a_0, \ldots, a_n</math> with integer coefficients, which is the determinant of the Sylvester matrix of Template:Math and Template:Math. The nonzero entries of the first column of the Sylvester matrix are <math>a_n</math> and <math>na_n,</math> and the resultant is thus a multiple of <math>a_n.</math> Hence the discriminant—up to its sign—is defined as the quotient of the resultant of Template:Math and Template:Math by <math>a_n</math>:

<math>\operatorname{Disc}_x(A) = \frac{(-1)^{n(n-1)/2}}{a_n} \operatorname{Res}_x(A,A')</math>

Historically, this sign has been chosen such that, over the reals, the discriminant will be positive when all the roots of the polynomial are real. The division by <math>a_n</math> may not be well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing <math>a_n</math> by 1 in the first column of the Sylvester matrix—before computing the determinant. In any case, the discriminant is a polynomial in <math>a_0, \ldots, a_n</math> with integer coefficients.

Expression in terms of the roots

When the above polynomial is defined over a field, it has Template:Math roots, <math>r_1, r_2, \dots, r_n</math>, not necessarily all distinct, in any algebraically closed extension of the field. (If the coefficients are real numbers, the roots may be taken in the field of complex numbers, where the fundamental theorem of algebra applies.)

In terms of the roots, the discriminant is equal to

<math>\operatorname{Disc}_x(A) = a_n^{2n-2}\prod_{i < j} (r_i-r_j)^2

= (-1)^{n(n-1)/2} a_n^{2n-2} \prod_{i \neq j} (r_i-r_j).</math>

It is thus the square of the Vandermonde polynomial times <math>a_n^{2n-2} </math>.

This expression for the discriminant is often taken as a definition. It makes clear that if the polynomial has a multiple root, then its discriminant is zero, and that, in the case of real coefficients, if all the roots are real and simple, then the discriminant is positive. Unlike the previous definition, this expression is not obviously a polynomial in the coefficients, but this follows either from the fundamental theorem of Galois theory, or from the fundamental theorem of symmetric polynomials and Vieta's formulas by noting that this expression is a symmetric polynomial in the roots of Template:Math.

Low degrees

The discriminant of a linear polynomial (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (using the usual conventions for the empty product and considering that one of the two blocks of the Sylvester matrix is empty). There is no common convention for the discriminant of a constant polynomial (i.e., polynomial of degree 0).

For small degrees, the discriminant is rather simple (see below), but for higher degrees, it may become unwieldy. For example, the discriminant of a general quartic has 16 terms,<ref>Template:Cite book </ref> that of a quintic has 59 terms,<ref>Template:Cite book </ref> and that of a sextic has 246 terms.<ref>Template:Cite book </ref> This is OEIS sequence A007878.

Degree 2

The quadratic polynomial <math>ax^2+bx+c \,</math> has discriminant

<math>b^2-4ac\,.</math>

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

<math>x_{1,2}=\frac{-b \pm \sqrt {b^2-4ac}}{2a}.</math>

where the discriminant is zero if and only if the two roots are equal. If Template:Math are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative.<ref>Template:Cite book</ref>

The discriminant is the product of Template:Math and the square of the difference of the roots.

If Template:Math are rational numbers, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.

Degree 3

File:Discriminant of cubic polynomials..png

The zero set of discriminant of the cubic Template:Math, i.e. points satisfying Template:Math.

The cubic polynomial <math>ax^3+bx^2+cx+d \,</math> has discriminant

<math>b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd\,.</math><ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In the special case of a depressed cubic polynomial <math>x^3+px+q</math>, the discriminant simplifies to

<math> -4p^3-27q^2\,.</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.<ref>Template:Cite book</ref>

The square root of a quantity strongly related to the discriminant appears in the formulas for the roots of a cubic polynomial. Specifically, this quantity can be Template:Math times the discriminant, or its product with the square of a rational number; for example, the square of Template:Math in the case of Cardano formula.

If the polynomial is irreducible and its coefficients are rational numbers (or belong to a number field), then the discriminant is a square of a rational number (or a number from the number field) if and only if the Galois group of the cubic equation is the cyclic group of order three.

Degree 4

File:Quartic Discriminant.png

The discriminant of the quartic polynomial Template:Math. The surface represents points (Template:Math) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.

The quartic polynomial <math> ax^4+bx^3+cx^2+dx+e\,</math> has discriminant

<math>\begin{align}

{} & 256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e \\[4pt] & {} -27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de \\[4pt] & {} +18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde \\[4pt] & {} -4b^3d^3-4b^2c^3e+b^2c^2d^2\,. \end{align}</math>

The depressed quartic polynomial <math> x^4+cx^2+dx+e\,</math> has discriminant

<math>\begin{align}

{} & 16c^4e -4c^3d^2 -128c^2e^2+144cd^2e -27d^4 + 256e^3\,. \end{align}</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.

Properties

Zero discriminant

The discriminant of a polynomial over a field is zero if and only if the polynomial has a multiple root in some field extension.

The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor.

In characteristic 0, this is equivalent to saying that the polynomial is not square-free (i.e., it is divisible by the square of a non-constant polynomial).

In nonzero characteristic Template:Math, the discriminant is zero if and only if the polynomial is not square-free or it has an irreducible factor which is not separable (i.e., the irreducible factor is a polynomial in <math>x^p</math>).

Invariance under change of the variable

The discriminant of a polynomial is, up to a scaling, invariant under any projective transformation of the variable. As a projective transformation may be decomposed into a product of translations, homotheties and inversions, this results in the following formulas for simpler transformations, where Template:Math denotes a polynomial of degree Template:Math, with <math>a_n</math> as leading coefficient.

• Invariance by translation:

<math>\operatorname{Disc}_x(P(x+\alpha)) = \operatorname{Disc}_x(P(x))</math>

This results from the expression of the discriminant in terms of the roots

• Invariance by homothety:

<math>\operatorname{Disc}_x(P(\alpha x)) = \alpha^{n(n-1)}\operatorname{Disc}_x(P(x))</math>

This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.

• Invariance by inversion:

<math>\operatorname{Disc}_x(P^{\mathrm{r}}\!\!\;(x)) = \operatorname{Disc}_x(P(x))</math>

when <math>P(0)\ne 0.</math> Here, <math>P^{\mathrm{r}}\!\!\;</math> denotes the reciprocal polynomial of Template:Math; that is, if <math>P(x) = a_nx^n + \cdots + a_0,</math> and <math>a_0 \neq 0,</math> then

<math>P^{\mathrm{r}}\!\!\;(x) = x^nP(1/x) = a_0x^n +\cdots +a_n.</math>

Invariance under ring homomorphisms

Let <math>\varphi\colon R \to S</math> be a homomorphism of commutative rings. Given a polynomial

<math>A = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0</math>

in Template:Math, the homomorphism <math>\varphi</math> acts on Template:Math for producing the polynomial

<math>A^\varphi = \varphi(a_n)x^n+\varphi(a_{n-1})x^{n-1}+ \cdots+\varphi(a_0)</math>

in Template:Math.

The discriminant is invariant under <math>\varphi</math> in the following sense. If <math>\varphi(a_n)\ne 0,</math> then

<math>\operatorname{Disc}_x(A^\varphi) = \varphi(\operatorname{Disc}_x(A)).</math>

As the discriminant is defined in terms of a determinant, this property results immediately from the similar property of determinants.

If <math>\varphi(a_n)= 0,</math> then <math>\varphi(\operatorname{Disc}_x(A))</math> may be zero or not. One has, when <math>\varphi(a_n)= 0,</math>

<math>\varphi(\operatorname{Disc}_x(A)) = \varphi(a_{n-1})^2\operatorname{Disc}_x(A^\varphi).</math>

When one is only interested in knowing whether a discriminant is zero (as is generally the case in algebraic geometry), these properties may be summarised as:

<math>\varphi(\operatorname{Disc}_x(A)) = 0</math> if and only if either <math>\operatorname{Disc}_x(A^\varphi)=0</math> or <math>\deg(A)-\deg(A^\varphi)\ge 2.</math>

This is often interpreted as saying that <math>\varphi(\operatorname{Disc}_x(A)) = 0</math> if and only if <math>A^\varphi</math> has a multiple root (possibly at infinity).

Product of polynomials

If Template:Math is a product of polynomials in Template:Math, then

<math>\begin{align}

\operatorname{disc}_x(R) &= \operatorname{disc}_x(P)\operatorname{Res}_x(P,Q)^2\operatorname{disc}_x(Q) \\[5pt] {}&=(-1)^{pq}\operatorname{disc}_x(P)\operatorname{Res}_x(P,Q)\operatorname{Res}_x(Q,P)\operatorname{disc}_x(Q), \end{align}</math> where <math>\operatorname{Res}_x</math> denotes the resultant with respect to the variable Template:Math, and Template:Math and Template:Math are the respective degrees of Template:Math and Template:Math.

This property follows immediately by substituting the expression for the resultant, and the discriminant, in terms of the roots of the respective polynomials.

Homogeneity

The discriminant is a homogeneous polynomial in the coefficients; it is also a homogeneous polynomial in the roots and thus quasi-homogeneous in the coefficients.

The discriminant of a polynomial of degree Template:Math is homogeneous of degree Template:Math in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by Template:Mvar does not change the roots, but multiplies the leading term by Template:Mvar. In terms of its expression as a determinant of a Template:Math matrix (the Sylvester matrix) divided by Template:Mvar, the determinant is homogeneous of degree Template:Math in the entries, and dividing by Template:Mvar makes the degree Template:Math.

The discriminant of a polynomial of degree Template:Math is homogeneous of degree Template:Math in the roots. This follows from the expression of the discriminant in terms of the roots, which is the product of a constant and <math>\binom{n}{2} = \frac{n(n-1)}{2}</math> squared differences of roots.

The discriminant of a polynomial of degree Template:Math is quasi-homogeneous of degree Template:Math in the coefficients, if, for every Template:Math, the coefficient of <math>x^i</math> is given the weight Template:Math. It is also quasi-homogeneous of the same degree, if, for every Template:Math, the coefficient of <math>x^i</math> is given the weight Template:Math. This is a consequence of the general fact that every polynomial which is homogeneous and symmetric in the roots may be expressed as a quasi-homogeneous polynomial in the elementary symmetric functions of the roots.

Consider the polynomial

<math> P=a_nx^n+a_{n-1}x^{n-1}+ \cdots +a_0.</math>

It follows from what precedes that the exponents in every monomial <math>a_0^{i_0}, \dots , a_n^{i_n}</math> appearing in the discriminant satisfy the two equations

<math>i_0+i_1+\cdots+i_n=2n-2</math>

and

<math>i_1+2i_2 + \cdots+n i_n=n(n-1),</math>

and also the equation

<math>ni_0 +(n-1)i_1+ \cdots+ i_{n-1}=n(n-1),</math>

which is obtained by subtracting the second equation from the first one multiplied by Template:Math.

This restricts the possible terms in the discriminant. For the general quadratic polynomial, the discriminant <math>b^2-4ac</math> is a homogeneous polynomial of degree 2 which has only two there are only two terms, while the general homogeneous polynomial of degree two in three variables has 6 terms. The discriminant of the general cubic polynomial is a homogeneous polynomial of degree 4 in four variables; it has five terms, which is the maximum allowed by the above rules, while the general homogeneous polynomial of degree 4 in 4 variables has 35 terms.

For higher degrees, there may be monomials which satisfy above rules and do not appear in the discriminant. The first example is for the quartic polynomial <math>ax^4 + bx^3 + cx^2 + dx + e</math>, in which case the monomial <math>bc^4d</math> satisfies the rules without appearing in the discriminant.

Real roots

In this section, all polynomials have real coefficients.

It has been seen in Template:Slink that the sign of the discriminant provides useful information on the nature of the roots for polynomials of degree 2 and 3. For higher degrees, the information provided by the discriminant is less complete, but still useful. More precisely, for a polynomial of degree Template:Math, one has:

• The polynomial has a multiple root if and only if its discriminant is zero.
• If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer Template:Math such that there are Template:Math pairs of complex conjugate roots and Template:Math real roots.
• If the discriminant is negative, the number of non-real roots is not a multiple of 4. That is, there is a nonnegative integer Template:Math such that there are Template:Math pairs of complex conjugate roots and Template:Math real roots.

Homogeneous bivariate polynomial

Let

<math>A(x,y) = a_0x^n+ a_1 x^{n-1}y + \cdots + a_n y^n=\sum_{i=0}^n a_i x^{n-i}y^i</math>

be a homogeneous polynomial of degree Template:Math in two indeterminates.

Supposing, for the moment, that <math>a_0</math> and <math>a_n</math> are both nonzero, one has

<math>\operatorname{Disc}_x(A(x,1))=\operatorname{Disc}_y(A(1,y)).</math>

Denoting this quantity by <math>\operatorname{Disc}^h (A),</math> one has

<math>\operatorname{Disc}_x (A) =y^{n(n-1)} \operatorname{Disc}^h (A),</math>

and

<math>\operatorname{Disc}_y (A) =x^{n(n-1)} \operatorname{Disc}^h (A).</math>

Because of these properties, the quantity <math>\operatorname{Disc}^h (A)</math> is called the discriminant or the homogeneous discriminant of Template:Math.

If <math>a_0</math> and <math>a_n</math> are permitted to be zero, the polynomials Template:Math and Template:Math may have a degree smaller than Template:Math. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree Template:Mvar. This means that the discriminants must be computed with <math>a_0</math> and <math>a_n</math> indeterminate, the substitution for them of their actual values being done after this computation. Equivalently, the formulas of Template:Slink must be used.

Use in algebraic geometry

The typical use of discriminants in algebraic geometry is for studying plane algebraic curves, and more generally algebraic hypersurfaces. Let Template:Math be such a curve or hypersurface; Template:Math is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface Template:Math in the space of the other indeterminates. The points of Template:Math are exactly the projection of the points of Template:Math (including the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected indeterminate.

For example, let Template:Mvar be a bivariate polynomial in Template:Mvar and Template:Mvar with real coefficients, so that Template:Math is the implicit equation of a real plane algebraic curve. Viewing Template:Mvar as a univariate polynomial in Template:Mvar with coefficients depending on Template:Mvar, then the discriminant is a polynomial in Template:Mvar whose roots are the Template:Mvar-coordinates of the singular points, of the points with a tangent parallel to the Template:Mvar-axis and of some of the asymptotes parallel to the Template:Mvar-axis. In other words, the computation of the roots of the Template:Mvar-discriminant and the Template:Mvar-discriminant allows one to compute all of the remarkable points of the curve, except the inflection points.

Generalizations

There are two classes of the concept of discriminant. The first class is the discriminant of an algebraic number field, which, in some cases including quadratic fields, is the discriminant of a polynomial defining the field.

Discriminants of the second class arise for problems depending on coefficients, when degenerate instances or singularities of the problem are characterized by the vanishing of a single polynomial in the coefficients. This is the case for the discriminant of a polynomial, which is zero when two roots collapse. Most of the cases, where such a generalized discriminant is defined, are instances of the following.

Let Template:Math be a homogeneous polynomial in Template:Math indeterminates over a field of characteristic 0, or of a prime characteristic that does not divide the degree of the polynomial. The polynomial Template:Math defines a projective hypersurface, which has singular points if and only the Template:Math partial derivatives of Template:Math have a nontrivial common zero. This is the case if and only if the multivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant of Template:Math. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of Template:Math, and it is better to take, as a discriminant, the primitive part of the resultant, computed with generic coefficients. The restriction on the characteristic is needed because otherwise a common zero of the partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials).

In the case of a homogeneous bivariate polynomial of degree Template:Math, this general discriminant is <math>d^{d-2}</math> times the discriminant defined in Template:Slink. Several other classical types of discriminants, that are instances of the general definition are described in next sections.

Quadratic forms

Template:See also A quadratic form is a function over a vector space, which is defined over some basis by a homogeneous polynomial of degree 2:

<math>Q(x_1,\ldots,x_n) \ =\ \sum_{i=1}^n a_{ii} x_i^2+\sum_{1\le i <j\le n}a_{ij}x_i x_j,</math>

or, in matrix form,

<math>Q(X) =X A X^\mathrm T,</math>

for the <math>n\times n</math> symmetric matrix <math>A=(a_{ij})</math>, the <math>1\times n</math> row vector <math>X=(x_1,\ldots,x_n)</math>, and the <math>n\times 1</math> column vector <math>X^{\mathrm{T}}</math>. In characteristic different from 2,<ref>In characteristic 2, the discriminant of a quadratic form is not defined, and is replaced by the Arf invariant.</ref> the discriminant or determinant of Template:Math is the determinant of Template:Math.<ref>Template:Cite book</ref>

The Hessian determinant of Template:Math is <math>2^n</math> times its discriminant. The multivariate resultant of the partial derivatives of Template:Math is equal to its Hessian determinant. So, the discriminant of a quadratic form is a special case of the above general definition of a discriminant.

The discriminant of a quadratic form is invariant under linear changes of variables (that is a change of basis of the vector space on which the quadratic form is defined) in the following sense: a linear change of variables is defined by a nonsingular matrix Template:Math, changes the matrix Template:Math into <math>S^\mathrm T A\,S,</math> and thus multiplies the discriminant by the square of the determinant of Template:Math. Thus the discriminant is well defined only up to the multiplication by a square. In other words, the discriminant of a quadratic form over a field Template:Math is an element of Template:Math, the quotient of the multiplicative monoid of Template:Math by the subgroup of the nonzero squares (that is, two elements of Template:Math are in the same equivalence class if one is the product of the other by a nonzero square). It follows that over the complex numbers, a discriminant is equivalent to 0 or 1. Over the real numbers, a discriminant is equivalent to −1, 0, or 1. Over the rational numbers, a discriminant is equivalent to a unique square-free integer.

By a theorem of Jacobi, a quadratic form over a field of characteristic different from 2 can be expressed, after a linear change of variables, in diagonal form as

<math>a_1x_1^2 + \cdots + a_nx_n^2.</math>

More precisely, a quadratic form may be expressed as a sum

<math>\sum_{i=1}^n a_i L_i^2</math>

where the Template:Math are independent linear forms and Template:Mvar is the number of the variables (some of the Template:Math may be zero). Equivalently, for any symmetric matrix Template:Math, there is an elementary matrix Template:Math such that <math>S^\mathrm T A\,S</math> is a diagonal matrix. Then the discriminant is the product of the Template:Math, which is well-defined as a class in Template:Math.

Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field).

A quadratic form in four variables is the equation of a projective surface. The surface has a singular point if and only its discriminant is zero. In this case, either the surface may be decomposed in planes, or it has a unique singular point, and is a cone or a cylinder. Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negative Gaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature.

Conic sections

A conic section is a plane curve defined by an implicit equation of the form

<math>ax^2+ 2bxy + cy^2 + 2dx + 2ey + f = 0,</math>

where Template:Math are real numbers.

Two quadratic forms, and thus two discriminants may be associated to a conic section.

The first quadratic form is

<math>ax^2+ 2bxy + cy^2 + 2dxz + 2eyz + fz^2 = 0.</math>

Its discriminant is the determinant

<math>\begin{vmatrix} a & b & d\\b & c & e\\d & e & f \end{vmatrix}. </math>

It is zero if the conic section degenerates into two lines, a double line or a single point.

The second discriminant, which is the only one that is considered in many elementary textbooks, is the discriminant of the homogeneous part of degree two of the equation. It is equal to<ref>Template:Cite book </ref>

<math>b^2 - ac,</math>

and determines the shape of the conic section. If this discriminant is negative, the curve either has no real points, or is an ellipse or a circle, or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is a parabola, or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is a hyperbola, or, if degenerated, a pair of intersecting lines.

Real quadric surfaces

A real quadric surface in the Euclidean space of dimension three is a surface that may be defined as the zeros of a polynomial of degree two in three variables. As for the conic sections there are two discriminants that may be naturally defined. Both are useful for getting information on the nature of a quadric surface.

Let <math>P(x,y,z)</math> be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, <math>Q_4,</math> depends on four variables, and is obtained by homogenizing Template:Math; that is

<math>Q_4(x,y,z,t)=t^2P(x/t,y/t, z/t).</math>

Let us denote its discriminant by <math>\Delta_4.</math>

The second quadratic form, <math>Q_3,</math> depends on three variables, and consists of the terms of degree two of Template:Math; that is

<math>Q_3(x,y,z)=Q_4(x, y,z,0).</math>

Let us denote its discriminant by <math>\Delta_3.</math>

If <math>\Delta_4>0,</math> and the surface has real points, it is either a hyperbolic paraboloid or a one-sheet hyperboloid. In both cases, this is a ruled surface that has a negative Gaussian curvature at every point.

If <math>\Delta_4<0,</math> the surface is either an ellipsoid or a two-sheet hyperboloid or an elliptic paraboloid. In all cases, it has a positive Gaussian curvature at every point.

If <math>\Delta_4=0,</math> the surface has a singular point, possibly at infinity. If there is only one singular point, the surface is a cylinder or a cone. If there are several singular points the surface consists of two planes, a double plane or a single line.

When <math>\Delta_4\ne 0,</math> the sign of <math>\Delta_3,</math> if not 0, does not provide any useful information, as changing Template:Math into Template:Math does not change the surface, but changes the sign of <math>\Delta_3.</math> However, if <math>\Delta_4\ne 0</math> and <math>\Delta_3 = 0,</math> the surface is a paraboloid, which is elliptic or hyperbolic, depending on the sign of <math>\Delta_4.</math>

Discriminant of an algebraic number field

Template:Main articleThe discriminant of an algebraic number field measures the size of the (ring of integers of the) algebraic number field.

More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. A theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.<ref>Template:Citation</ref>

Let K be an algebraic number field, and let O_K be its ring of integers. Let b₁, ..., b_n be an integral basis of O_K (i.e. a basis as a Z-module), and let {σ₁, ..., σ_n} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σ_i(b_j). Symbolically,

<math>\Delta_K=\det\left(\begin{array}{cccc}

\sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\ \sigma_2(b_1) & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ \sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n) \end{array}\right)^2. </math>

The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.

Fundamental discriminants

A specific type of discriminant useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integral binary quadratic forms, which are expressions of the form:<math display="block">Q(x, y) = ax^2 + bxy + cy^2</math>

where <math display="inline">a</math>, <math display="inline">b</math>, and <math display="inline">c</math> are integers. The discriminant of <math display="inline">Q(x, y)</math> is given by:<math display="block">D = b^2 - 4ac</math>Not every integer can arise as a discriminant of an integral binary quadratic form. An integer <math display="inline">D</math> is a fundamental discriminant if and only if it meets one of the following criteria:

• Case 1: <math display="inline">D</math> is congruent to 1 modulo 4 (<math display="inline">D \equiv 1 \pmod{4}</math>) and is square-free, meaning it is not divisible by the square of any prime number.
• Case 2: <math display="inline">D</math> is equal to four times an integer <math display="inline">m</math> (<math display="inline">D = 4m</math>) where <math display="inline">m</math> is congruent to 2 or 3 modulo 4 (<math display="inline">m \equiv 2, 3 \pmod{4}</math>) and is square-free.

These conditions ensure that every fundamental discriminant corresponds uniquely to a specific type of quadratic form.

The first eleven positive fundamental discriminants are:

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequence A003658 in the OEIS).

The first eleven negative fundamental discriminants are:

−3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in the OEIS).

Quadratic number fields

A quadratic field is a field extension of the rational numbers <math display="inline">\mathbb{Q}</math> that has degree 2. The discriminant of a quadratic field plays a role analogous to the discriminant of a quadratic form.

There exists a fundamental connection: an integer <math display="inline">D_0</math> is a fundamental discriminant if and only if:

• <math display="inline">D_0 = 1</math>, or
• <math display="inline">D_0</math> is the discriminant of a quadratic field.

For each fundamental discriminant <math display="inline">D_0 \neq 1</math>, there exists a unique (up to isomorphism) quadratic field with <math display="inline">D_0</math> as its discriminant. This connects the theory of quadratic forms and the study of quadratic fields.

Prime factorization

Fundamental discriminants can also be characterized by their prime factorization. Consider the set <math display="inline">S</math> consisting of <math>-8, 8, -4,</math> the prime numbers congruent to 1 modulo 4, and the additive inverses of the prime numbers congruent to 3 modulo 4:<math display="block">S = \{-8, -4, 8, -3, 5, -7, -11, 13, 17, -19, ... \}</math>An integer <math display="inline">D \neq 1</math> is a fundamental discriminant if and only if it is a product of elements of <math>S</math> that are pairwise coprime.Template:Cn

External links

• Wolfram Mathworld: Polynomial Discriminant
• Planetmath: Discriminant