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Discriminant

From Wikipedia, the free encyclopedia
Function of the coefficients of a polynomial that gives information on its roots
Not to be confused withDeterminant.
For other uses, seeDiscriminant (disambiguation).
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Inmathematics, thediscriminant of apolynomial is a quantity that depends on thecoefficients and allows deducing some properties of theroots without computing them. More precisely, it is apolynomial function of the coefficients of the original polynomial. The discriminant is widely used inpolynomial factoring,number theory, andalgebraic geometry.

The discriminant of thequadratic polynomialax2+bx+c{\displaystyle ax^{2}+bx+c} is

b2−4ac,{\displaystyle b^{2}-4ac,}

the quantity which appears under thesquare root in thequadratic formula. Ifa≠0,{\displaystyle a\neq 0,} this discriminant is zeroif and only if the polynomial has adouble root. In the case ofreal coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinctcomplex conjugate roots.[1] Similarly, the discriminant of acubic polynomial is zero if and only if the polynomial has amultiple 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 positivedegree 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 amultiple of 4 (including none), and negative otherwise.

Several generalizations are also called discriminant: thediscriminant of an algebraic number field; thediscriminant of aquadratic form; and more generally, thediscriminant of aform, of ahomogeneous polynomial, or of aprojective hypersurface (these three concepts are essentially equivalent).

Origin

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The term "discriminant" was coined in 1851 by the British mathematicianJames Joseph Sylvester.[2]

Definition

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Let

A(x)=anxn+an−1xn−1+⋯+a1x+a0{\displaystyle A(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}

be a polynomial ofdegreen (this meansan≠0{\displaystyle a_{n}\neq 0}), such that the coefficientsa0,…,an{\displaystyle a_{0},\ldots ,a_{n}} belong to afield, or, more generally, to acommutative ring. Theresultant ofA with itsderivative,

A′(x)=nanxn−1+(n−1)an−1xn−2+⋯+a1,{\displaystyle A'(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots +a_{1},}

is a polynomial ina0,…,an{\displaystyle a_{0},\ldots ,a_{n}} withinteger coefficients, which is thedeterminant of theSylvester matrix ofA andA′. The nonzero entries of the first column of the Sylvester matrix arean{\displaystyle a_{n}} andnan,{\displaystyle na_{n},} and theresultant is thus a multiple ofan.{\displaystyle a_{n}.} Hence the discriminant—up to its sign—is defined as the quotient of the resultant ofA andA' byan{\displaystyle a_{n}}:

Discx⁡(A)=(−1)n(n−1)/2anResx⁡(A,A′){\displaystyle \operatorname {Disc} _{x}(A)={\frac {(-1)^{n(n-1)/2}}{a_{n}}}\operatorname {Res} _{x}(A,A')}

where the resultant is computed withA′{\displaystyle A'} considered of degreen−1,{\displaystyle n-1,} even if it has a lower degree, which occurs when the characteristic dividesn{\displaystyle n}.

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 byan{\displaystyle a_{n}} may not be well defined if thering of the coefficients containszero divisors. Such a problem may be avoided by replacingan{\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 ina0,…,an{\displaystyle a_{0},\ldots ,a_{n}} with integer coefficients.

Expression in terms of the roots

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When the above polynomial is defined over afield, it hasn roots,r1,r2,…,rn{\displaystyle r_{1},r_{2},\dots ,r_{n}}, not necessarily all distinct, in anyalgebraically closed extension of the field. (If the coefficients are real numbers, the roots may be taken in the field ofcomplex numbers, where thefundamental theorem of algebra applies.)

In terms of the roots, the discriminant is equal to

Discx⁡(A)=an2n−2∏i<j(ri−rj)2=(−1)n(n−1)/2an2n−2∏i≠j(ri−rj).{\displaystyle \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}).}

It is thus the square of theVandermonde polynomial timesan2n−2{\displaystyle a_{n}^{2n-2}}.

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

Low degrees

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The discriminant of alinear polynomial (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (using the usual conventions for theempty product and considering that one of the two blocks of theSylvester matrix isempty). 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 ageneralquartic has 16 terms,[3] that of aquintic has 59 terms,[4] and that of asextic has 246 terms.[5]This isOEIS sequenceA007878.

Degree 2

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See also:Quadratic equation § Discriminant

The quadratic polynomialax2+bx+c{\displaystyle ax^{2}+bx+c\,} has discriminant

b2−4ac.{\displaystyle b^{2}-4ac\,.}

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

x1,2=−b±b2−4ac2a.{\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. Ifa,b,c are real numbers, the polynomial has two distinct real roots if the discriminant is positive, and twocomplex conjugate roots if it is negative.[6]

The discriminant is the product ofa2 and the square of the difference of the roots.

Ifa,b,c arerational numbers, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.

Degree 3

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See also:Cubic equation § Discriminant
The zero set of discriminant of the cubicx3 +bx2 +cx +d, i.e. points satisfyingb2c2 − 4c3 − 4b3d − 27d2 + 18bcd = 0.

The cubic polynomialax3+bx2+cx+d{\displaystyle ax^{3}+bx^{2}+cx+d\,} has discriminant

b2c2−4ac3−4b3d−27a2d2+18abcd.{\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}[7][8]

In the special case of adepressed cubic polynomialx3+px+q{\displaystyle x^{3}+px+q}, the discriminant simplifies to

−4p3−27q2.{\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.[9]

The square root of a quantity strongly related to the discriminant appears in theformulas 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 of1/18 in the case ofCardano formula.

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

Degree 4

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The discriminant of the quartic polynomialx4 +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.

Thequartic polynomialax4+bx3+cx2+dx+e{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,}has discriminant

256a3e3−192a2bde2−128a2c2e2+144a2cd2e−27a2d4+144ab2ce2−6ab2d2e−80abc2de+18abcd3+16ac4e−4ac3d2−27b4e2+18b3cde−4b3d3−4b2c3e+b2c2d2.{\displaystyle {\begin{aligned}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\[4pt]&\quad -27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\[4pt]&\quad +18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\[4pt]&\quad -4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}.\end{aligned}}}

The depressed quartic polynomialx4+cx2+dx+e{\displaystyle x^{4}+cx^{2}+dx+e\,}has discriminant

16c4e−4c3d2−128c2e2+144cd2e−27d4+256e3.{\displaystyle {\begin{aligned}{}&16c^{4}e-4c^{3}d^{2}-128c^{2}e^{2}+144cd^{2}e-27d^{4}+256e^{3}.\end{aligned}}}

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

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Zero discriminant

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

The discriminant of a polynomial over anintegral domain is zero if and only if the polynomial and itsderivative have a non-constant common divisor.

Incharacteristic 0, this is equivalent to saying that the polynomial is notsquare-free (i.e., it is divisible by the square of a non-constant polynomial).

In nonzero characteristicp, the discriminant is zero if and only if the polynomial is not square-free or it has anirreducible factor which is not separable (i.e., the irreducible factor is a polynomial inxp{\displaystyle x^{p}}).

Invariance under change of the variable

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The discriminant of a polynomial is,up to a scaling, invariant under anyprojective 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, whereP(x) denotes a polynomial of degreen, withan{\displaystyle a_{n}} as leading coefficient.

  • Invariance by translation:
Discx⁡(P(x+α))=Discx⁡(P(x)){\displaystyle \operatorname {Disc} _{x}(P(x+\alpha ))=\operatorname {Disc} _{x}(P(x))}
This results from the expression of the discriminant in terms of the roots
  • Invariance by homothety:
Discx⁡(P(αx))=αn(n−1)Discx⁡(P(x))Discx⁡(αP(x))=α2n−2Discx⁡(P(x)){\displaystyle {\begin{aligned}\operatorname {Disc} _{x}(P(\alpha x))&=\alpha ^{n(n-1)}\operatorname {Disc} _{x}(P(x))\\\operatorname {Disc} _{x}(\alpha P(x))&=\alpha ^{2n-2}\operatorname {Disc} _{x}(P(x))\end{aligned}}}
This results from the expression in terms of the roots, or of the quasi-homogeneity of the discriminant.
  • Invariance by inversion:
Discx⁡(Pr(x))=Discx⁡(P(x)){\displaystyle \operatorname {Disc} _{x}(P^{\mathrm {r} }\!\!\;(x))=\operatorname {Disc} _{x}(P(x))}
whenP(0)≠0.{\displaystyle P(0)\neq 0.} Here,Pr{\displaystyle P^{\mathrm {r} }\!\!\;} denotes thereciprocal polynomial ofP; that is, ifP(x)=anxn+⋯+a0,{\displaystyle P(x)=a_{n}x^{n}+\cdots +a_{0},} anda0≠0,{\displaystyle a_{0}\neq 0,} then
Pr(x)=xnP(1/x)=a0xn+⋯+an.{\displaystyle P^{\mathrm {r} }\!\!\;(x)=x^{n}P(1/x)=a_{0}x^{n}+\cdots +a_{n}.}

Invariance under ring homomorphisms

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Letφ:R→S{\displaystyle \varphi \colon R\to S} be ahomomorphism ofcommutative rings. Given a polynomial

A=anxn+an−1xn−1+⋯+a0{\displaystyle A=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}}

inR[x], the homomorphismφ{\displaystyle \varphi } acts onA for producing the polynomial

Aφ=φ(an)xn+φ(an−1)xn−1+⋯+φ(a0){\displaystyle A^{\varphi }=\varphi (a_{n})x^{n}+\varphi (a_{n-1})x^{n-1}+\cdots +\varphi (a_{0})}

inS[x].

The discriminant is invariant underφ{\displaystyle \varphi } in the following sense. Ifφ(an)≠0,{\displaystyle \varphi (a_{n})\neq 0,} then

Discx⁡(Aφ)=φ(Discx⁡(A)).{\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=\varphi (\operatorname {Disc} _{x}(A)).}

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

Ifφ(an)=0,{\displaystyle \varphi (a_{n})=0,} thenφ(Discx⁡(A)){\displaystyle \varphi (\operatorname {Disc} _{x}(A))} may be zero or not. One has, whenφ(an)=0,{\displaystyle \varphi (a_{n})=0,}

φ(Discx⁡(A))=φ(an−1)2Discx⁡(Aφ).{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=\varphi (a_{n-1})^{2}\operatorname {Disc} _{x}(A^{\varphi }).}

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

φ(Discx⁡(A))=0{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0} if and only if eitherDiscx⁡(Aφ)=0{\displaystyle \operatorname {Disc} _{x}(A^{\varphi })=0} ordeg⁡(A)−deg⁡(Aφ)≥2.{\displaystyle \deg(A)-\deg(A^{\varphi })\geq 2.}

This is often interpreted as saying thatφ(Discx⁡(A))=0{\displaystyle \varphi (\operatorname {Disc} _{x}(A))=0} if and only ifAφ{\displaystyle A^{\varphi }} has amultiple root (possiblyat infinity).

Product of polynomials

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IfR =PQ is a product of polynomials inx, then

discx⁡(R)=discx⁡(P)Resx⁡(P,Q)2discx⁡(Q)=(−1)pqdiscx⁡(P)Resx⁡(P,Q)Resx⁡(Q,P)discx⁡(Q),{\displaystyle {\begin{aligned}\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{aligned}}}

whereResx{\displaystyle \operatorname {Res} _{x}} denotes theresultant with respect to the variablex, andp andq are the respective degrees ofP andQ.

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

Homogeneity

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

The discriminant of a polynomial of degreen is homogeneous of degree2n − 2 in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients byλ does not change the roots, but multiplies the leading term byλ. In terms of its expression as a determinant of a(2n − 1) × (2n − 1)matrix (theSylvester matrix) divided byan, the determinant is homogeneous of degree2n − 1 in the entries, and dividing byan makes the degree2n − 2.

The discriminant of a polynomial of degreen is homogeneous of degreen(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(n2)=n(n−1)2{\displaystyle {\binom {n}{2}}={\frac {n(n-1)}{2}}} squared differences of roots.

The discriminant of a polynomial of degreen is quasi-homogeneous of degreen(n − 1) in the coefficients, if, for everyi, the coefficient ofxi{\displaystyle x^{i}} is given the weightn −i. It is also quasi-homogeneous of the same degree, if, for everyi, the coefficient ofxi{\displaystyle x^{i}} is given the weighti. This is a consequence of the general fact that every polynomial which is homogeneous andsymmetric in the roots may be expressed as a quasi-homogeneous polynomial in theelementary symmetric functions of the roots.

Consider the polynomial

P=anxn+an−1xn−1+⋯+a0.{\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}.}

It follows from what precedes that the exponents in everymonomiala0i0,…,anin{\displaystyle a_{0}^{i_{0}},\dots ,a_{n}^{i_{n}}} appearing in the discriminant satisfy the two equations

i0+i1+⋯+in=2n−2{\displaystyle i_{0}+i_{1}+\cdots +i_{n}=2n-2}

and

i1+2i2+⋯+nin=n(n−1),{\displaystyle i_{1}+2i_{2}+\cdots +ni_{n}=n(n-1),}

and also the equation

ni0+(n−1)i1+⋯+in−1=n(n−1),{\displaystyle ni_{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 byn.

This restricts the possible terms in the discriminant. For the general quadratic polynomial, the discriminantb2−4ac{\displaystyle b^{2}-4ac} 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 polynomialax4+bx3+cx2+dx+e{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e}, in which case the monomialbc4d{\displaystyle bc^{4}d} satisfies the rules without appearing in the discriminant.

Real roots

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In this section, all polynomials havereal coefficients.

It has been seen in§ Low degrees 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 degreen, one has:

  • The polynomial has amultiple 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 integerk ≤n/4 such that there are2k pairs ofcomplex conjugate roots andn − 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 integerk ≤ (n − 2)/4 such that there are2k + 1 pairs of complex conjugate roots andn − 4k + 2 real roots.

Homogeneous bivariate polynomial

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Let

A(x,y)=a0xn+a1xn−1y+⋯+anyn=∑i=0naixn−iyi{\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 ahomogeneous polynomial of degreen in two indeterminates.

Supposing, for the moment, thata0{\displaystyle a_{0}} andan{\displaystyle a_{n}} are both nonzero, one has

Discx⁡(A(x,1))=Discy⁡(A(1,y)).{\displaystyle \operatorname {Disc} _{x}(A(x,1))=\operatorname {Disc} _{y}(A(1,y)).}

Denoting this quantity byDisch⁡(A),{\displaystyle \operatorname {Disc} ^{h}(A),}one has

Discx⁡(A)=yn(n−1)Disch⁡(A),{\displaystyle \operatorname {Disc} _{x}(A)=y^{n(n-1)}\operatorname {Disc} ^{h}(A),}

and

Discy⁡(A)=xn(n−1)Disch⁡(A).{\displaystyle \operatorname {Disc} _{y}(A)=x^{n(n-1)}\operatorname {Disc} ^{h}(A).}

Because of these properties, the quantityDisch⁡(A){\displaystyle \operatorname {Disc} ^{h}(A)} is called thediscriminant or thehomogeneous discriminant ofA.

Ifa0{\displaystyle a_{0}} andan{\displaystyle a_{n}} are permitted to be zero, the polynomialsA(x, 1) andA(1,y) may have a degree smaller thann. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degreen. This means that the discriminants must be computed witha0{\displaystyle a_{0}} andan{\displaystyle a_{n}} indeterminate, the substitution for them of their actual values being doneafter this computation. Equivalently, the formulas of§ Invariance under ring homomorphisms must be used.

Use in algebraic geometry

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The typical use of discriminants inalgebraic geometry is for studying planealgebraic curves, and more generallyalgebraic hypersurfaces. LetV be such a curve or hypersurface;V is defined as the zero set of amultivariate 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 hypersurfaceW in the space of the other indeterminates. The points ofW are exactly the projection of the points ofV (including thepoints at infinity), which either are singular or have atangent hyperplane that is parallel to the axis of the selected indeterminate.

For example, letf be a bivariate polynomial inX andY with real coefficients, so that f  = 0 is the implicit equation of a real planealgebraic curve. Viewingf as a univariate polynomial inY with coefficients depending onX, then the discriminant is a polynomial inX whose roots are theX-coordinates of the singular points, of the points with a tangent parallel to theY-axis and of some of the asymptotes parallel to theY-axis. In other words, the computation of the roots of theY-discriminant and theX-discriminant allows one to compute all of the remarkable points of the curve, except theinflection points.

Generalizations

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There are two classes of the concept of discriminant. The first class is thediscriminant of an algebraic number field, which, in some cases includingquadratic 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.

LetA be a homogeneous polynomial inn indeterminates over a field ofcharacteristic 0, or of aprime characteristic that does notdivide the degree of the polynomial. The polynomialA defines aprojective hypersurface, which hassingular points if and only thenpartial derivatives ofA have a nontrivial commonzero. This is the case if and only if themultivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant ofA. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power ofn, and it is better to take, as a discriminant, theprimitive 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 (seeEuler's identity for homogeneous polynomials).

In the case of a homogeneous bivariate polynomial of degreed, this general discriminant isdd−2{\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

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See also:Fundamental discriminant

Aquadratic form is a function over avector space, which is defined over somebasis by ahomogeneous polynomial of degree 2:

Q(x1,…,xn) = ∑i=1naiixi2+∑1≤i<j≤naijxixj,{\displaystyle Q(x_{1},\ldots ,x_{n})\ =\ \sum _{i=1}^{n}a_{ii}x_{i}^{2}+\sum _{1\leq i<j\leq n}a_{ij}x_{i}x_{j},}

or, in matrix form,

Q(X)=XAXT,{\displaystyle Q(X)=XAX^{\mathrm {T} },}

for then×n{\displaystyle n\times n}symmetric matrixA=(aij){\displaystyle A=(a_{ij})}, the1×n{\displaystyle 1\times n} row vectorX=(x1,…,xn){\displaystyle X=(x_{1},\ldots ,x_{n})}, and then×1{\displaystyle n\times 1} column vectorXT{\displaystyle X^{\mathrm {T} }}. Incharacteristic different from 2,[10] thediscriminant ordeterminant ofQ is thedeterminant ofA.[11]

TheHessian determinant ofQ is2n{\displaystyle 2^{n}} times its discriminant. Themultivariate resultant of the partial derivatives ofQ 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 achange 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 anonsingular matrixS, changes the matrixA intoSTAS,{\displaystyle S^{\mathrm {T} }A\,S,} and thus multiplies the discriminant by the square of the determinant ofS. Thus the discriminant is well defined onlyup to the multiplication by a square. In other words, the discriminant of a quadratic form over a fieldK is an element ofK/(K×)2, thequotient of the multiplicativemonoid ofK by thesubgroup of the nonzero squares (that is, two elements ofK are in the sameequivalence class if one is the product of the other by a nonzero square). It follows that over thecomplex numbers, a discriminant is equivalent to 0 or 1. Over thereal numbers, a discriminant is equivalent to −1, 0, or 1. Over therational numbers, a discriminant is equivalent to a uniquesquare-free integer.

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

a1x12+⋯+anxn2.{\displaystyle a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}.}

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

∑i=1naiLi2{\displaystyle \sum _{i=1}^{n}a_{i}L_{i}^{2}}

where theLi are independent linear forms andn is the number of the variables (some of theai may be zero). Equivalently, for any symmetric matrixA, there is anelementary matrixS such thatSTAS{\displaystyle S^{\mathrm {T} }A\,S} is adiagonal matrix.Then the discriminant is the product of theai, which is well-defined as a class inK/(K×)2.

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

A quadratic form in four variables is the equation of aprojective surface. The surface has asingular 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 acone or acylinder. Over the reals, if the discriminant is positive, then the surface either has no real point or has everywhere a negativeGaussian curvature. If the discriminant is negative, the surface has real points, and has a negative Gaussian curvature.

Conic sections

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Aconic section is aplane curve defined by animplicit equation of the form

ax2+2bxy+cy2+2dx+2ey+f=0,{\displaystyle ax^{2}+2bxy+cy^{2}+2dx+2ey+f=0,}

wherea,b,c,d,e,f are real numbers.

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

The first quadratic form is

ax2+2bxy+cy2+2dxz+2eyz+fz2=0.{\displaystyle ax^{2}+2bxy+cy^{2}+2dxz+2eyz+fz^{2}=0.}

Its discriminant is thedeterminant

|abdbcedef|.{\displaystyle {\begin{vmatrix}a&b&d\\b&c&e\\d&e&f\end{vmatrix}}.}

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[12]

b2−ac,{\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 anellipse or acircle, or, if degenerated, is reduced to a single point. If the discriminant is zero, the curve is aparabola, or, if degenerated, a double line or two parallel lines. If the discriminant is positive, the curve is ahyperbola, or, if degenerated, a pair of intersecting lines.

Real quadric surfaces

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A realquadric surface in theEuclidean 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.

LetP(x,y,z){\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,Q4,{\displaystyle Q_{4},} depends on four variables, and is obtained byhomogenizingP; that is

Q4(x,y,z,t)=t2P(x/t,y/t,z/t).{\displaystyle Q_{4}(x,y,z,t)=t^{2}P(x/t,y/t,z/t).}

Let us denote its discriminant byΔ4.{\displaystyle \Delta _{4}.}

The second quadratic form,Q3,{\displaystyle Q_{3},} depends on three variables, and consists of the terms of degree two ofP; that is

Q3(x,y,z)=Q4(x,y,z,0).{\displaystyle Q_{3}(x,y,z)=Q_{4}(x,y,z,0).}

Let us denote its discriminant byΔ3.{\displaystyle \Delta _{3}.}

IfΔ4>0,{\displaystyle \Delta _{4}>0,} and the surface has real points, it is either ahyperbolic paraboloid or aone-sheet hyperboloid. In both cases, this is aruled surface that has a negativeGaussian curvature at every point.

IfΔ4<0,{\displaystyle \Delta _{4}<0,} the surface is either anellipsoid or atwo-sheet hyperboloid or anelliptic paraboloid. In all cases, it has a positiveGaussian curvature at every point.

IfΔ4=0,{\displaystyle \Delta _{4}=0,} the surface has asingular point, possiblyat infinity. If there is only one singular point, the surface is acylinder or acone. If there are several singular points the surface consists of two planes, a double plane or a single line.

WhenΔ4≠0,{\displaystyle \Delta _{4}\neq 0,} the sign ofΔ3,{\displaystyle \Delta _{3},} if not 0, does not provide any useful information, as changingP into−P does not change the surface, but changes the sign ofΔ3.{\displaystyle \Delta _{3}.} However, ifΔ4≠0{\displaystyle \Delta _{4}\neq 0} andΔ3=0,{\displaystyle \Delta _{3}=0,} the surface is aparaboloid, which is elliptic or hyperbolic, depending on the sign ofΔ4.{\displaystyle \Delta _{4}.}

Discriminant of an algebraic number field

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Main article:Discriminant of an algebraic number field

The discriminant of analgebraic 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 thefundamental domain of thering of integers, and it regulates whichprimes areramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several importantanalytic formulas such as thefunctional equation of theDedekind zeta function ofK, and theanalytic class number formula forK.A theorem ofHermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still anopen problem, and the subject of current research.[13]

LetK be an algebraic number field, and letOK be itsring of integers. Letb1, ...,bn be anintegral basis ofOK (i.e. a basis as aZ-module), and let {σ1, ..., σn} be the set of embeddings ofK into thecomplex numbers (i.e.injectivering homomorphismsK → C). Thediscriminant ofK is thesquare of thedeterminant of then bynmatrixB whose (i,j)-entry is σi(bj). Symbolically,

ΔK=det(σ1(b1)σ1(b2)⋯σ1(bn)σ2(b1)⋱⋮⋮⋱⋮σn(b1)⋯⋯σn(bn))2.{\displaystyle \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}.}


The discriminant ofK can be referred to as the absolute discriminant ofK to distinguish it from the of anextensionK/L of number fields. The latter is anideal in the ring of integers ofL, and like the absolute discriminant it indicates which primes are ramified inK/L. It is a generalization of the absolute discriminant allowing forL to be bigger thanQ; in fact, whenL = Q, the relative discriminant ofK/Q is theprincipal ideal ofZ generated by the absolute discriminant ofK.

Fundamental discriminants

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A specific type of discriminant useful in the study of quadratic fields is the fundamental discriminant. It arises in the theory of integralbinary quadratic forms, which are expressions of the form:Q(x,y)=ax2+bxy+cy2{\displaystyle Q(x,y)=ax^{2}+bxy+cy^{2}}


wherea{\textstyle a},b{\textstyle b}, andc{\textstyle c} are integers. The discriminant ofQ(x,y){\textstyle Q(x,y)} is given by:D=b2−4ac{\displaystyle D=b^{2}-4ac}Not every integer can arise as a discriminant of an integral binary quadratic form. An integerD{\textstyle D} is a fundamental discriminant if and only if it meets one of the following criteria:

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 theOEIS).

The first eleven negative fundamental discriminants are:

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

Quadratic number fields

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A quadratic field is a field extension of the rational numbersQ{\textstyle \mathbb {Q} } 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 integerD0{\textstyle D_{0}} is a fundamental discriminant if and only if:

For each fundamental discriminantD0≠1{\textstyle D_{0}\neq 1}, there exists a unique (up to isomorphism) quadratic field withD0{\textstyle D_{0}} as its discriminant. This connects the theory of quadratic forms and the study of quadratic fields.

Prime factorization

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Fundamental discriminants can also be characterized by their prime factorization. Consider the setS{\textstyle S} consisting of−8,8,−4,{\displaystyle -8,8,-4,} the prime numbers congruent to 1 modulo 4, and theadditive inverses of the prime numbers congruent to 3 modulo 4:S={−8,−4,8,−3,5,−7,−11,13,17,−19,...}{\displaystyle S=\{-8,-4,8,-3,5,-7,-11,13,17,-19,...\}}An integerD≠1{\textstyle D\neq 1} is a fundamental discriminant if and only if it is a product of elements ofS{\displaystyle S} that are pairwisecoprime.[citation needed]

References

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  1. ^"Discriminant | mathematics".Encyclopedia Britannica. Retrieved2020-08-09.
  2. ^Sylvester, J. J. (1851). "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants".Philosophical Magazine. 4th series.2:391–410.
    Sylvester coins the word "discriminant" onpage 406.
  3. ^Wang, Dongming (2004).Elimination practice: software tools and applications.Imperial College Press. ch. 10 p. 180.ISBN 1-86094-438-8.
  4. ^Gelfand, Israel M.;Kapranov, Mikhail M.;Zelevinsky, Andrei V. (1994).Discriminants, resultants and multidimensional determinants.Birkhäuser. p. 1.ISBN 3-7643-3660-9. Archived fromthe original on 2013-01-13.
  5. ^Dickenstein, Alicia; Emiris, Ioannis Z. (2005).Solving polynomial equations: foundations, algorithms, and applications.Springer. ch. 1 p. 26.ISBN 3-540-24326-7.
  6. ^Irving, Ronald S. (2004).Integers, polynomials, and rings. Springer-Verlag New York, Inc. ch. 10.3 pp. 153–154.ISBN 0-387-40397-3.
  7. ^"Cubic Discriminant | Brilliant Math & Science Wiki". Retrieved2023-03-21.
  8. ^"Discriminant of a cubic equation". 14 July 2019. Retrieved2023-03-21.
  9. ^Irving, Ronald S. (2004).Integers, polynomials, and rings. Springer-Verlag New York, Inc. ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156.ISBN 0-387-40397-3.
  10. ^In characteristic 2, the discriminant of a quadratic form is not defined, and is replaced by theArf invariant.
  11. ^Cassels, J. W. S. (1978).Rational Quadratic Forms. London Mathematical Society Monographs. Vol. 13.Academic Press. p. 6.ISBN 0-12-163260-1.Zbl 0395.10029.
  12. ^Fanchi, John R. (2006).Math refresher for scientists and engineers. John Wiley and Sons. sec. 3.2, p. 45.ISBN 0-471-75715-2.
  13. ^Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel (2002), "A Survey of Discriminant Counting", in Fieker, Claus; Kohel, David R. (eds.),Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002, Lecture Notes in Computer Science, vol. 2369, Berlin: Springer-Verlag, pp. 80–94,doi:10.1007/3-540-45455-1_7,ISBN 978-3-540-43863-2,ISSN 0302-9743,MR 2041075

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