Discriminant

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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 ${\displaystyle a{x}^2+bx+c}$ is

${\displaystyle b^2-4ac,}$

the quantity which appears under the square root in the quadratic formula. If ${\displaystyle a\ne 0,}$ 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. 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.

Definition

Let

${\displaystyle A(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0}$

be a polynomial of degree n (this means ${\displaystyle a_n\ne 0}$), such that the coefficients ${\displaystyle a_0,\dots ,a_n}$ belong to a field, or, more generally, to a commutative ring. The resultant of A and its derivative,

${\displaystyle A^{\prime }(x)=n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\cdots +a_1,}$

is a polynomial in ${\displaystyle a_0,\dots ,a_n}$ with integer coefficients, which is the determinant of the Sylvester matrix of A and A′. The nonzero entries of the first column of the Sylvester matrix are ${\displaystyle a_n}$ and ${\displaystyle n{a}_n,}$ and the resultant is thus a multiple of ${\displaystyle a_n.}$ Hence the discriminant—up to its sign—is defined as the quotient of the resultant of A and A' by ${\displaystyle a_n}$:

${\displaystyle {\mathrm{Disc}}_x(A)=\frac{(-1{)}^{n(n-1)/2}}{a_n}{\mathrm{Res}}_x(A,A^{\prime })}$

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 ${\displaystyle a_n}$ may not be well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing ${\displaystyle a_n}$ by 1 in the first column of the Sylvester matrix—before computing the determinant. In any case, the discriminant is a polynomial in ${\displaystyle a_0,\dots ,a_n}$ with integer coefficients.

Expression in terms of the roots

When the above polynomial is defined over a field, it has n roots, ${\displaystyle r_1,r_2,\dots ,r_n}$, 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

${\displaystyle {\mathrm{Disc}}_x(A)=a_n^{2n-2}\prod _{i<j}(r_i-r_j{)}^2=(-1{)}^{n(n-1)/2}{a}_n^{2n-2}\prod _{i\ne j}(r_i-r_j).}$

It is thus the square of the Vandermonde polynomial times ${\displaystyle a_n^{2n-2}}$.

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 A.

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, that of a quintic has 59 terms, and that of a sextic has 246 terms. This is OEIS sequence A007878.

Degree 2

See also: Quadratic equation § Discriminant

The quadratic polynomial ${\displaystyle a{x}^2+bx+c}$ has discriminant

${\displaystyle b^2-4ac.}$

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

${\displaystyle x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.}$

where the discriminant is zero if and only if the two roots are equal. If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and two complex conjugate roots if it is negative.

The discriminant is the product of a² and the square of the difference of the roots.

If a, b, c 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

See also: Cubic equation § Discriminant

 

The zero set of discriminant of the cubic x³ + bx² + cx + d, i.e. points satisfying b²c² – 4c³ – 4b³d – 27d² + 18bcd = 0.

The cubic polynomial ${\displaystyle a{x}^3+b{x}^2+cx+d}$ has discriminant

${\displaystyle b^2{c}^2-4a{c}^3-4b^{3}d-27a^2{d}^2+18abcd.}$

In the special case of a depressed cubic polynomial ${\displaystyle x^3+px+q}$, the discriminant simplifies to

${\displaystyle -4p^3-27q^2.}$

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.

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 −3 times the discriminant, or its product with the square of a rational number; for example, the square of 1/18 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

The discriminant of the quartic polynomial x⁴ + cx² + dx + e. The surface represents points (c, d, e) 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 ${\displaystyle a{x}^4+b{x}^3+c{x}^2+dx+e}$ has discriminant

${\displaystyle \begin{array}{rl}& 256a^3{e}^3-192a^{2}bd{e}^2-128a^2{c}^2{e}^2+144a^{2}c{d}^{2}e\\ & -27a^2{d}^4+144a{b}^{2}c{e}^2-6a{b}^2{d}^{2}e-80ab{c}^{2}de\\ & +18abc{d}^3+16a{c}^{4}e-4a{c}^3{d}^2-27b^4{e}^2+18b^{3}cde\\ & -4b^3{d}^3-4b^2{c}^{3}e+b^2{c}^2{d}^2.\end{array}}$

The depressed quartic polynomial ${\displaystyle x^4+c{x}^2+dx+e}$ has discriminant

${\displaystyle \begin{array}{rl}& 16c^{4}e-4c^3{d}^2-128c^2{e}^2+144c{d}^{2}e-27d^4+256e^3.\end{array}}$

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., divisible by the square of a non-constant polynomial).

In nonzero characteristic p, 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 ${\displaystyle x^p}$).

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 P(x) denotes a polynomial of degree n, with ${\displaystyle a_n}$ as leading coefficient.

• Invariance by translation:

${\displaystyle {\mathrm{Disc}}_x(P(x+\alpha ))={\mathrm{Disc}}_x(P(x))}$

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

• Invariance by homothety:

${\displaystyle {\mathrm{Disc}}_x(P(\alpha x))={\alpha }^{n(n-1)}{\mathrm{Disc}}_x(P(x))}$

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

• Invariance by inversion:

${\displaystyle {\mathrm{Disc}}_x(P^{\mathrm{r}}(x))={\mathrm{Disc}}_x(P(x))}$

when ${\displaystyle P(0)\ne 0.}$ Here, ${\displaystyle P^{\mathrm{r}}}$ denotes the reciprocal polynomial of P; that is, if ${\displaystyle P(x)=a_n{x}^n+\cdots +a_0,}$ and ${\displaystyle a_0\ne 0,}$ then

${\displaystyle P^{\mathrm{r}}(x)=x^{n}P(1/x)=a_0{x}^n+\cdots +a_n.}$

Invariance under ring homomorphisms

Let ${\displaystyle \phi :R\to S}$ be a homomorphism of commutative rings. Given a polynomial

${\displaystyle A=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0}$

in R[x], the homomorphism ${\displaystyle \phi }$ acts on A for producing the polynomial

${\displaystyle A^{\phi }=\phi (a_n)x^n+\phi (a_{n-1})x^{n-1}+\cdots +\phi (a_0)}$

in S[x].

The discriminant is invariant under ${\displaystyle \phi }$ in the following sense. If ${\displaystyle \phi (a_n)\ne 0,}$ then

${\displaystyle {\mathrm{Disc}}_x(A^{\phi })=\phi ({\mathrm{Disc}}_x(A)).}$

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

If ${\displaystyle \phi (a_n)=0,}$ then ${\displaystyle \phi ({\mathrm{Disc}}_x(A))}$ may be zero or not. One has, when ${\displaystyle \phi (a_n)=0,}$

${\displaystyle \phi ({\mathrm{Disc}}_x(A))=\phi (a_{n-1}{)}^2{\mathrm{Disc}}_x(A^{\phi }).}$

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:

${\displaystyle \phi ({\mathrm{Disc}}_x(A))=0}$ if and only if either ${\displaystyle {\mathrm{Disc}}_x(A^{\phi })=0}$ or ${\displaystyle \mathrm{deg}(A)-\mathrm{deg}(A^{\phi })\ge 2.}$

This is often interpreted as saying that ${\displaystyle \phi ({\mathrm{Disc}}_x(A))=0}$ if and only if ${\displaystyle A^{\phi }}$ has a multiple root (possibly at infinity).

Product of polynomials

If R = PQ is a product of polynomials in x, then

${\displaystyle \begin{array}{rl}{\mathrm{disc}}_x(R)& ={\mathrm{disc}}_x(P){\mathrm{Res}}_x(P,Q{)}^2{\mathrm{disc}}_x(Q)\\ & =(-1{)}^{pq}{\mathrm{disc}}_x(P){\mathrm{Res}}_x(P,Q){\mathrm{Res}}_x(Q,P){\mathrm{disc}}_x(Q),\end{array}}$

where ${\displaystyle {\mathrm{Res}}_x}$ denotes the resultant with respect to the variable x, and p and q are the respective degrees of P and Q.

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 n is homogeneous of degree 2n − 2 in the coefficients. This can be seen two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by λ does not change the roots, but multiplies the leading term by λ. In terms of its expression as a determinant of a (2n − 1) × (2n − 1) matrix (the Sylvester matrix) divided by a_n, the determinant is homogeneous of degree 2n − 1 in the entries, and dividing by a_n makes the degree 2n − 2.

The discriminant of a polynomial of degree n is homogeneous of degree n(n − 1) in the roots. This follows from the expression of the discriminant in terms of the roots, which is the product of a constant and ${\displaystyle (\genfrac{}{}{0ex}{}{n}{2})=\frac{n(n-1)}{2}}$ squared differences of roots.

The discriminant of a polynomial of degree n is quasi-homogeneous of degree n(n − 1) in the coefficients, if, for every i, the coefficient of ${\displaystyle x^i}$ is given the weight n − i. It is also quasi-homogeneous of the same degree, if, for every i, the coefficient of ${\displaystyle x^i}$ is given the weight i. 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

${\displaystyle P=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0.}$

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

${\displaystyle i_0+i_1+\cdots +i_n=2n-2}$

and

${\displaystyle i_1+2i_2+\cdots +n{i}_n=n(n-1),}$

and also the equation

${\displaystyle n{i}_0+(n-1)i_1+\cdots +i_{n-1}=n(n-1),}$

which is obtained by subtracting the second equation from the first one multiplied by n.

This restricts the possible terms in the discriminant. For the general quadratic polynomial there are only two possibilities and two terms in the discriminant, while the general homogeneous polynomial of degree two in three variables has 6 terms. For the general cubic polynomial, there are five possibilities and five terms in the discriminant, while the general homogeneous polynomial of degree 4 in 5 variables has 70 terms.

For higher degrees, there may be monomials which satisfy above equations and do not appear in the discriminant. The first example is for the quartic polynomial ${\displaystyle a{x}^4+b{x}^3+c{x}^2+dx+e}$, in which case the monomial ${\displaystyle b{c}^{4}d}$ satisfies the equations without appearing in the discriminant.

Real roots

In this section, all polynomials have real coefficients.

It has been seen in § Low degrees that the sign of the discriminant provides a full 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 n, 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 k ≤ n/4 such that there are 2k pairs of complex conjugate roots and n − 4k 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 k ≤ (n − 2)/4 such that there are 2k + 1 pairs of complex conjugate roots and n − 4k + 2 real roots.

Homogeneous bivariate polynomial

Let

${\displaystyle A(x,y)=a_0{x}^n+a_1{x}^{n-1}y+\cdots +a_n{y}^n=\sum _{i=0}^n{a}_i{x}^{n-i}{y}^i}$

be a homogeneous polynomial of degree n in two indeterminates.

Supposing, for the moment, that ${\displaystyle a_0}$ and ${\displaystyle a_n}$ are both nonzero, one has

${\displaystyle {\mathrm{Disc}}_x(A(x,1))={\mathrm{Disc}}_y(A(1,y)).}$

Denoting this quantity by ${\displaystyle {\mathrm{Disc}}^h(A),}$ one has

${\displaystyle {\mathrm{Disc}}_x(A)=y^{n(n-1)}{\mathrm{Disc}}^h(A),}$

and

${\displaystyle {\mathrm{Disc}}_y(A)=x^{n(n-1)}{\mathrm{Disc}}^h(A).}$

Because of these properties, the quantity ${\displaystyle {\mathrm{Disc}}^h(A)}$ is called the discriminant or the homogeneous discriminant of A.

If ${\displaystyle a_0}$ and ${\displaystyle a_n}$ are permitted to be zero, the polynomials A(x, 1) and A(1, y) may have a degree smaller than n. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree n. This means that the discriminants must be computed with ${\displaystyle a_0}$ and ${\displaystyle a_n}$ indeterminate, the substitution for them of their actual values being done after this computation. Equivalently, the formulas of § Invariance under ring homomorphisms 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 V be such a curve or hypersurface; V 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 W in the space of the other indeterminates. The points of W are exactly the projection of the points of V (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 f be a bivariate polynomial in X and Y with real coefficients, so that f  = 0 is the implicit equation of a real plane algebraic curve. Viewing f as a univariate polynomial in Y with coefficients depending on X, then the discriminant is a polynomial in X whose roots are the X-coordinates of the singular points, of the points with a tangent parallel to the Y-axis and of some of the asymptotes parallel to the Y-axis. In other words, the computation of the roots of the Y-discriminant and the X-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 A be a homogeneous polynomial in n indeterminates over a field of characteristic 0, or of a prime characteristic that does not divide the degree of the polynomial. The polynomial A defines a projective hypersurface, which has singular points if and only the n partial derivatives of A 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 A. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of n, 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 d, this general discriminant is ${\displaystyle d^{d-2}}$ times the discriminant defined in § Homogeneous bivariate polynomial. Several other classical types of discriminants, that are instances of the general definition are described in next sections.

Quadratic forms

See also: Fundamental discriminant

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

${\displaystyle Q(x_1,\dots ,x_n)=\sum _{i=1}^n{a}_{ii}{x}_i^2+\sum _{1\le i<j\le n}{a}_{ij}{x}_i{x}_j,}$

or, in matrix form,

${\displaystyle Q(X)=XA{X}^{\mathrm{T}},}$

for the ${\displaystyle n\times n}$ symmetric matrix ${\displaystyle A=(a_{ij})}$, the ${\displaystyle 1\times n}$ row vector ${\displaystyle X=(x_1,\dots ,x_n)}$, and the ${\displaystyle n\times 1}$ column vector ${\displaystyle X^{\mathrm{T}}}$. In characteristic different from 2,^[8] the discriminant or determinant of Q is the determinant of A.^[9]

The Hessian determinant of Q is ${\displaystyle 2^n}$ times its discriminant. The multivariate resultant of the partial derivatives of Q 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 S, changes the matrix A into ${\displaystyle S^{\mathrm{T}}AS,}$ and thus multiplies the discriminant by the square of the determinant of S. 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 K is an element of K/(K^×)², the quotient of the multiplicative monoid of K by the subgroup of the nonzero squares (that is, two elements of K 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

${\displaystyle a_1{x}_1^2+\cdots +a_n{x}_n^2.}$

More precisely, a quadratic forms on may be expressed as a sum

${\displaystyle \sum _{i=1}^n{a}_i{L}_i^2}$

where the L_i are independent linear forms and n is the number of the variables (some of the a_i may be zero). Equivalently, for any symmetric matrix A, there is an elementary matrix S such that ${\displaystyle S^{\mathrm{T}}AS}$ is a diagonal matrix. Then the discriminant is the product of the a_i, which is well-defined as a class in K/(K^×)².

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

${\displaystyle a{x}^2+2bxy+c{y}^2+2dx+2ey+f=0,}$

where a, b, c, d, e, f are real numbers.

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

The first quadratic form is

${\displaystyle a{x}^2+2bxy+c{y}^2+2dxz+2eyz+f{z}^2=0.}$

Its discriminant is the determinant

${\displaystyle \left|\begin{array}{ccc}a& b& d\\ b& c& e\\ d& e& f\end{array}\right|.}$

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

${\displaystyle b^2-ac,}$

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 ${\displaystyle P(x,y,z)}$ be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, ${\displaystyle Q_4,}$ depends on four variables, and is obtained by homogenizing P; that is

${\displaystyle Q_4(x,y,z,t)=t^{2}P(x/t,y/t,z/t).}$

Let us denote its discriminant by ${\displaystyle {\mathrm{\Delta }}_4.}$

The second quadratic form, ${\displaystyle Q_3,}$ depends on three variables, and consists of the terms of degree two of P; that is

${\displaystyle Q_3(x,y,z)=Q_4(x,y,z,0).}$

Let us denote its discriminant by ${\displaystyle {\mathrm{\Delta }}_3.}$

If ${\displaystyle {\mathrm{\Delta }}_4>0,}$ 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 ${\displaystyle {\mathrm{\Delta }}_4<0,}$ 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 ${\displaystyle {\mathrm{\Delta }}_4=0,}$ 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 ${\displaystyle {\mathrm{\Delta }}_4\ne 0,}$ the sign of ${\displaystyle {\mathrm{\Delta }}_3,}$ if not 0, does not provide any useful information, as changing P into −P does not change the surface, but changes the sign of ${\displaystyle {\mathrm{\Delta }}_3.}$ However, if ${\displaystyle {\mathrm{\Delta }}_4\ne 0}$ and ${\displaystyle {\mathrm{\Delta }}_3=0,}$ the surface is a paraboloid, which is elliptic or hyperbolic, depending on the sign of ${\displaystyle {\mathrm{\Delta }}_4.}$

Causal sets

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Causal_sets

The program is based on a theorem by David Malament that states that if there is a bijective map between two past and future distinguishing space times that preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume of regions in the spacetime. This volume factor can be recovered by specifying a volume element for each space time point. The volume of a space time region could then be found by counting the number of points in that region.

Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the program. He has coined the slogan "Order + Number = Geometry" to characterize the above argument. The program provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance.

Definition

A causal set (or causet) is a set ${\displaystyle C}$ with a partial order relation ${\displaystyle ⪯}$ that is

• Reflexive: For all ${\displaystyle x\in C}$, we have ${\displaystyle x⪯x}$.
• Antisymmetric: For all ${\displaystyle x,y\in C}$, we have ${\displaystyle x⪯y}$ and ${\displaystyle y⪯x}$ implies ${\displaystyle x=y}$.
• Transitive: For all ${\displaystyle x,y,z\in C}$, we have ${\displaystyle x⪯y}$ and ${\displaystyle y⪯z}$ implies ${\displaystyle x⪯z}$.
• Locally finite: For all ${\displaystyle x,z\in C}$, we have ${\displaystyle \{y\in C|x⪯y⪯z\}}$ is a finite set.

We'll write ${\displaystyle x\prec y}$ if ${\displaystyle x⪯y}$ and ${\displaystyle x\ne y}$.

The set ${\displaystyle C}$ represents the set of spacetime events and the order relation ${\displaystyle ⪯}$ represents the causal relationship between events (see causal structure for the analogous idea in a Lorentzian manifold).

Although this definition uses the reflexive convention we could have chosen the irreflexive convention in which the order relation is irreflexive and asymmetric.

The causal relation of a Lorentzian manifold (without closed causal curves) satisfies the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

Comparison to the continuum

Given a causal set we may ask whether it can be embedded into a Lorentzian manifold. An embedding would be a map taking elements of the causal set into points in the manifold such that the order relation of the causal set matches the causal ordering of the manifold. A further criterion is needed however before the embedding is suitable. If, on average, the number of causal set elements mapped into a region of the manifold is proportional to the volume of the region then the embedding is said to be faithful. In this case we can consider the causal set to be 'manifold-like'.

A central conjecture of the causal set program, called the Hauptvermutung ('fundamental conjecture'), now refuted, is that the same causal set cannot be faithfully embedded into two spacetimes that are not similar on large scales.

It is difficult to define this conjecture precisely because it is difficult to decide when two spacetimes are 'similar on large scales'.

Modelling spacetime as a causal set would require us to restrict attention to those causal sets that are 'manifold-like'. Given a causal set this is a difficult property to determine.

Regardless, the Hauptvermutung has now been shown to be false.

Sprinkling

A plot of 1000 sprinkled points in 1+1 dimensions

The difficulty of determining whether a causal set can be embedded into a manifold can be approached from the other direction. We can create a causal set by sprinkling points into a Lorentzian manifold. By sprinkling points in proportion to the volume of the spacetime regions and using the causal order relations in the manifold to induce order relations between the sprinkled points, we can produce a causal set that (by construction) can be faithfully embedded into the manifold.

To maintain Lorentz invariance this sprinkling of points must be done randomly using a Poisson process. Thus the probability of sprinkling ${\displaystyle n}$ points into a region of volume ${\displaystyle V}$ is

${\displaystyle P(n)=\frac{(\rho V{)}^n{e}^{-\rho V}}{n!}}$

where ${\displaystyle \rho }$ is the density of the sprinkling.

Sprinkling points as a regular lattice would not keep the number of points proportional to the region volume.

Geometry

Some geometrical constructions in manifolds carry over to causal sets. When defining these we must remember to rely only on the causal set itself, not on any background spacetime into which it might be embedded. For an overview of these constructions, see.

Geodesics

A plot of geodesics between two points in a 180-point causal set made by sprinkling into 1+1 dimensions

A link in a causal set is a pair of elements ${\displaystyle x,y\in C}$ such that ${\displaystyle x\prec y}$ but with no ${\displaystyle z\in C}$ such that ${\displaystyle x\prec z\prec y}$.

A chain is a sequence of elements ${\displaystyle x_0,x_1,\dots ,x_n}$ such that ${\displaystyle x_i\prec x_{i+1}}$ for ${\displaystyle i=0,\dots ,n-1}$. The length of a chain is ${\displaystyle n}$. If every ${\displaystyle x_i,x_{i+1}}$ in the chain form a link, then the chain is called a path.

We can use this to define the notion of a geodesic between two causal set elements, provided they are order comparable, that is, causally connected (physically, this means they are time-like). A geodesic between two elements ${\displaystyle x⪯y\in C}$ is a chain consisting only of links such that

1. ${\displaystyle x_0=x}$ and ${\displaystyle x_n=y}$
2. The length of the chain, ${\displaystyle n}$, is maximal over all chains from ${\displaystyle x}$ to ${\displaystyle y}$.

In general there can be more than one geodesic between two comparable elements.

Myrheim first suggested that the length of such a geodesic should be directly proportional to the proper time along a timelike geodesic joining the two spacetime points. Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes. The proportionality has been shown to hold and is conjectured to hold for sprinklings in curved spacetimes too.

Dimension estimators

Much work has been done in estimating the manifold dimension of a causal set. This involves algorithms using the causal set aiming to give the dimension of the manifold into which it can be faithfully embedded. The algorithms developed so far are based on finding the dimension of a Minkowski spacetime into which the causal set can be faithfully embedded.

• Myrheim–Meyer dimension

This approach relies on estimating the number of ${\displaystyle k}$-length chains present in a sprinkling into ${\displaystyle d}$-dimensional Minkowski spacetime. Counting the number of ${\displaystyle k}$-length chains in the causal set then allows an estimate for ${\displaystyle d}$ to be made.

• Midpoint-scaling dimension

This approach relies on the relationship between the proper time between two points in Minkowski spacetime and the volume of the spacetime interval between them. By computing the maximal chain length (to estimate the proper time) between two points ${\displaystyle x}$ and ${\displaystyle y}$ and counting the number of elements ${\displaystyle z}$ such that ${\displaystyle x\prec z\prec y}$ (to estimate the volume of the spacetime interval) the dimension of the spacetime can be calculated.

These estimators should give the correct dimension for causal sets generated by high-density sprinklings into ${\displaystyle d}$-dimensional Minkowski spacetime. Tests in conformally-flat spacetimes^[4] have shown these two methods to be accurate.

Dynamics

An ongoing task is to develop the correct dynamics for causal sets. These would provide a set of rules that determine which causal sets correspond to physically realistic spacetimes. The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics. This approach would perform a "sum-over-causal sets" by growing a causal set one element at a time. Elements would be added according to quantum mechanical rules and interference would ensure a large manifold-like spacetime would dominate the contributions. The best model for dynamics at the moment is a classical model in which elements are added according to probabilities. This model, due to David Rideout and Rafael Sorkin, is known as classical sequential growth (CSG) dynamics. The classical sequential growth model is a way to generate causal sets by adding new elements one after another. Rules for how new elements are added are specified and, depending on the parameters in the model, different causal sets result.

In analogy to the path integral formulation of quantum mechanics, one approach to developing a quantum dynamics for causal sets has been to apply an action principle in the sum-over-causal sets approach. Sorkin has proposed a discrete analogue for the d'Alembertian, which can in turn be used to define the Ricci curvature scalar and thereby the Benincasa–Dowker action on a causal set. Monte-Carlo simulations have provided evidence for a continuum phase in 2D using the Benincasa–Dowker action.