Editing Discriminant

{{short description|Function of the coefficients of a polynomial that gives information on its roots}}
{{distinguish|Determinant}}
{{Other uses}}
{{more citations needed|date=November 2011}}

In [[mathematics]], the '''discriminant''' of a [[polynomial]] is a quantity that depends on the [[coefficient]]s and allows deducing some properties of the [[zero of a function|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 factorization|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 number|real]] coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct [[complex conjugate]] roots.<ref>{{Cite web|title=Discriminant {{!}} mathematics|url=https://www.britannica.com/science/discriminant|access-date=2020-08-09|website=Encyclopedia Britannica|language=en}}</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 of a polynomial|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 (mathematics)|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 (mathematics)|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>{{cite journal|first=J. J.|last=Sylvester|date=1851|title=On a remarkable discovery in the theory of canonical forms and of hyperdeterminants|journal=Philosophical Magazine|series=4th series|volume=2|pages=391–410}}<br>Sylvester coins the word "discriminant" on [https://books.google.com/books?id=DBNDAQAAIAAJ&pg=PA406 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 of a polynomial|degree]] {{math|''n''}} (this means <math>a_n\ne 0</math>), such that the coefficients <math>a_0, \ldots, a_n</math> belong to a [[field (mathematics)|field]], or, more generally, to a [[commutative ring]]. The [[resultant]] of {{math|''A''}} with its [[formal derivative|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 {{math|''A''}} and {{math|''A''{{void}}′}}. 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 {{math|''A''}} and {{math|''A'{{void}}''}} 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 (mathematics)|ring]] of the coefficients contains [[zero divisor]]s. 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 (mathematics)|field]], it has {{math|''n''}} 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 number]]s, 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 root|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 {{math|''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 matrix|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 [[generic polynomial|general]] [[Quartic function|quartic]] has 16 terms,<ref>{{cite book
|title=Elimination practice: software tools and applications
|first1=Dongming
|last1=Wang
|publisher=[[Imperial College Press]]
|year=2004
|isbn=1-86094-438-8
|url=https://books.google.com/books?id=ucpk6oO5GN0C&pg=PA180
|at=ch. 10 p. 180}}
</ref> that of a [[Quintic function|quintic]] has 59 terms,<ref>{{cite book
|title=Discriminants, resultants and multidimensional determinants
|first1=Israel M.
|last1=Gelfand
|author1-link=Israel Gelfand
|first2=Mikhail M.
|last2=Kapranov
|author2-link=Mikhail Kapranov
|first3=Andrei V.
|last3=Zelevinsky
|author3-link=Andrei Zelevinsky
|publisher=[[Birkhäuser]]
|year=1994
|isbn=3-7643-3660-9
|page=1
|url=http://blms.oxfordjournals.org/cgi/reprint/28/1/96|archive-url=https://archive.today/20130113052223/http://blms.oxfordjournals.org/cgi/reprint/28/1/96|url-status=dead|archive-date=2013-01-13}}
</ref> and that of a [[sextic equation|sextic]] has 246 terms.<ref>{{cite book
|title=Solving polynomial equations: foundations, algorithms, and applications
|first1=Alicia
|last1=Dickenstein
|author1-link=Alicia Dickenstein
|first2=Ioannis Z.
|last2=Emiris
|publisher=[[Springer Science+Business Media|Springer]]
|year=2005
|isbn=3-540-24326-7
|url=https://books.google.com/books?id=rSs-pQNrO_YC&pg=PA26
|at=ch. 1 p. 26}}
</ref>
This is [[OEIS]] sequence {{OEIS link|A007878}}.
<!-- Please don't add numbers of terms of higher degrees (like 7/1103, 8/5247 and others of http://oeis.org/A007878) without providing proper sources. Thanks -->

===Degree 2===
{{see also|Quadratic equation#Discriminant}}

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 {{math|''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.<ref>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=0-387-40397-3
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=ch. 10.3 pp. 153–154}}</ref>

The discriminant is the product of {{math|''a''{{sup|2}}}} and the square of the difference of the roots.

If {{math|''a'', ''b'', ''c''}} are [[rational number]]s, then the discriminant is the square of a rational number if and only if the two roots are rational numbers.

===Degree 3===
{{seealso|Cubic equation#Discriminant}}
[[File:Discriminant of cubic polynomials..png|thumb|The zero set of discriminant of the cubic {{math|''x''<sup>3</sup> + ''bx''<sup>2</sup> + ''cx'' + ''d''}}, i.e. points satisfying {{math|1=''b''<sup>2</sup>''c''<sup>2</sup> − 4''c''<sup>3</sup> − 4''b''<sup>3</sup>''d'' − 27''d''<sup>2</sup> + 18''bcd'' = 0}}.]]
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>{{cite web| url=https://brilliant.org/wiki/cubic-discriminant/| title=Cubic Discriminant {{!}} Brilliant Math & Science Wiki| access-date=2023-03-21}}</ref><ref>{{cite web| url=https://www.johndcook.com/blog/2019/07/14/discriminant-of-a-cubic/| title=Discriminant of a cubic equation| date=14 July 2019| access-date=2023-03-21}}</ref>

In the special case of a [[Depressed cubic#Depressed cubic|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>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=0-387-40397-3
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156}}</ref>

The square root of a quantity strongly related to the discriminant appears in the [[Cubic equation#General cubic formula|formulas for the roots of a cubic polynomial]]. Specifically, this quantity can be {{math|−3}} times the discriminant, or its product with the square of a rational number; for example, the square of {{math|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 (group theory)|order]] three.

===Degree 4===
[[File:Quartic Discriminant.png|thumb|The discriminant of the quartic polynomial {{math|''x''<sup>4</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e''}}. The surface represents points ({{math|''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]]
<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 (mathematics)|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 [[formal derivative|derivative]] have a non-constant common divisor.

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

In nonzero characteristic {{math|''p''}}, the discriminant is zero if and only if the polynomial is not square-free or it has an [[irreducible polynomial|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 {{math|''P''(''x'')}} denotes a polynomial of degree {{math|''n''}}, 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 {{math|''P''}}; 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 [[ring homomorphism|homomorphism]] of [[commutative ring]]s. Given a polynomial 
:<math>A = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0</math>
in {{math|''R''[''x'']}}, the homomorphism <math>\varphi</math> acts on {{math|''A''}} 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 {{math|''S''[''x'']}}.

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 [[point at infinity|at infinity]]).

===Product of polynomials===
If {{math|1=''R'' = ''PQ''}} is a product of polynomials in {{math|''x''}}, 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 {{math|''x''}}, and {{math|''p''}} and {{math|''q''}} are the respective degrees of {{math|''P''}} and {{math|''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 polynomial|quasi-homogeneous]] in the coefficients.

The discriminant of a polynomial of degree {{math|''n''}} is homogeneous of degree {{math|2''n'' − 2}} in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by {{mvar|λ}} does not change the roots, but multiplies the leading term by {{mvar|λ}}. In terms of its expression as a determinant of a {{math|(2''n'' − 1)&thinsp;×&thinsp;(2''n'' − 1)}} [[matrix (mathematics)|matrix]] (the [[Sylvester matrix]]) divided by {{mvar|a<sub>n</sub>}}, the determinant is homogeneous of degree {{math|2''n'' − 1}} in the entries, and dividing by {{mvar|a<sub>n</sub>}} makes the degree {{math|2''n'' − 2}}.

The discriminant of a polynomial of degree {{math|''n''}} is homogeneous of degree {{math|''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 <math>\binom{n}{2} = \frac{n(n-1)}{2}</math> squared differences of roots.

The discriminant of a polynomial of degree {{math|''n''}} is quasi-homogeneous of degree {{math|''n''(''n'' − 1)}} in the coefficients, if, for every {{math|''i''}}, the coefficient of <math>x^i</math> is given the weight {{math|''n'' − ''i''}}. It is also quasi-homogeneous of the same degree, if, for every {{math|''i''}}, the coefficient of <math>x^i</math> is given the weight {{math|''i''}}. This is a consequence of the general fact that every polynomial which is homogeneous and [[symmetric polynomial|symmetric]] in the roots may be expressed as a quasi-homogeneous polynomial in the [[elementary symmetric function]]s 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 {{math|''n''}}.

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 number|real]] coefficients.

It has been seen in {{slink||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 degree {{math|''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 {{math|''k'' ≤ ''n''/4}} such that there are {{math|2''k''}} pairs of [[complex conjugate]] roots and {{math|''n'' − 4''k''}} 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 {{math|''k'' ≤ (''n'' − 2)/4}} such that there are {{math|2''k'' + 1}} pairs of complex conjugate roots and {{math|''n'' − 4''k'' + 2}} 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 {{math|''n''}} 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 {{math|''A''}}.

If <math>a_0</math> and <math>a_n</math> are permitted to be zero, the polynomials {{math|''A''(''x'', 1)}} and {{math|''A''(1, ''y'')}} may have a degree smaller than {{math|''n''}}. In this case, above formulas and definition remain valid, if the discriminants are computed as if all polynomials would have the degree {{mvar|''n''}}. 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 {{slink||Invariance under ring homomorphisms}} must be used.

==Use in algebraic geometry==

The typical use of discriminants in [[algebraic geometry]] is for studying plane [[algebraic curve]]s, and more generally [[Hypersurface|algebraic hypersurface]]s. Let {{math|''V''}} be such a curve or hypersurface; {{math|''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 {{math|''W''}} in the space of the other indeterminates. The points of {{math|''W''}} are exactly the projection of the points of {{math|''V''}} (including the [[points at infinity]]), which either are singular or have a [[tangent space|tangent hyperplane]] that is parallel to the axis of the selected indeterminate.

For example, let {{mvar|f}} be a bivariate polynomial in {{mvar|X}} and {{mvar|Y}} with real coefficients, so that&nbsp;{{math|1=''f'' &thinsp;= 0}} is the implicit equation of a real plane [[algebraic curve]]. Viewing {{mvar|f}} as a univariate polynomial in {{mvar|Y}} with coefficients depending on {{mvar|X}}, then the discriminant is a polynomial in {{mvar|X}} whose roots are the {{mvar|X}}-coordinates of the singular points, of the points with a tangent parallel to the {{mvar|Y}}-axis and of some of the asymptotes parallel to the {{mvar|Y}}-axis. In other words, the computation of the roots of the {{mvar|Y}}-discriminant and the {{mvar|X}}-discriminant allows one to compute all of the remarkable points of the curve, except the [[inflection point]]s.

==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 field]]s, 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 {{math|''A''}} be a homogeneous polynomial in {{math|''n''}} indeterminates over a field of [[characteristic (algebra)|characteristic]] 0, or of a [[prime number|prime]] characteristic that does not [[divisor|divide]] the degree of the polynomial. The polynomial {{math|''A''}} defines a [[projective hypersurface]], which has [[singular point of an algebraic variety|singular points]] if and only the {{math|''n''}} [[partial derivative]]s of {{math|''A''}} have a nontrivial common [[zero of a function|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 {{math|''A''}}. However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of {{math|''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 {{math|''d''}}, this general discriminant is <math>d^{d-2}</math> times the discriminant defined in {{slink||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 (vector space)|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 (algebra)|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 {{math|''Q''}} is the [[determinant]] of {{math|''A''}}.<ref>{{cite book | first=J. W. S. | last=Cassels | author-link=J. W. S. Cassels | title=Rational Quadratic Forms | series=London Mathematical Society Monographs | volume=13 | publisher=[[Academic Press]] | year=1978 | isbn=0-12-163260-1 | zbl=0395.10029 | page=6 }}</ref>

The [[Hessian determinant]] of {{math|''Q''}} is <math>2^n</math> times its discriminant. The [[multivariate resultant]] of the partial derivatives of {{math|''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]] {{math|''S''}}, changes the matrix {{math|''A''}} into <math>S^\mathrm T A\,S,</math> and thus multiplies the discriminant by the square of the determinant of {{math|''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 {{math|''K''}} is an element of {{math|''K''/(''K''<sup>×</sup>)<sup>2</sup>}}, the [[quotient monoid|quotient]] of the multiplicative [[monoid]] of {{math|''K''}} by the [[subgroup]] of the nonzero squares (that is, two elements of {{math|''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 number]]s, a discriminant is equivalent to 0 or 1. Over the [[real number]]s, a discriminant is equivalent to −1, 0, or 1. Over the [[rational number]]s, a discriminant is equivalent to a unique [[square-free integer]].

By a theorem of [[Carl Gustav Jacob Jacobi|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 {{math|''L''<sub>''i''</sub>}} are independent linear forms and {{mvar|n}} is the number of the variables (some of the {{math|''a''<sub>''i''</sub>}} may be zero). Equivalently, for any symmetric matrix {{math|''A''}}, there is an [[elementary matrix]] {{math|''S''}} such that <math>S^\mathrm T A\,S</math> is a [[diagonal matrix]].
Then the discriminant is the product of the {{math|''a''<sub>''i''</sub>}}, which is well-defined as a class in {{math|''K''/(''K''<sup>×</sup>)<sup>2</sup>}}.

Geometrically, the discriminant of a quadratic form in three variables is the equation of a [[projective curve|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 of an algebraic variety|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 {{math|''a'', ''b'', ''c'', ''d'', ''e'', ''f''}} are real numbers.

Two [[quadratic form]]s, 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>{{cite book
|title=Math refresher for scientists and engineers
|first1=John R.
|last1=Fanchi
|publisher=John Wiley and Sons
|year=2006
|isbn=0-471-75715-2
|url=https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45
|at=sec. 3.2, p. 45}}
</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 [[homogenization of a polynomial|homogenizing]] {{math|''P''}}; 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 {{math|''P''}}; 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 of an algebraic variety|singular point]], possibly [[point at infinity|at infinity]]. If there is only one singular point, the surface is a [[cylinder]] or a [[conic surface|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 {{math|''P''}} into {{math|−''P''}} 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===
{{main article|Discriminant of an algebraic number field}}The 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 [[Prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].

The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[Functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. [[Hermite–Minkowski theorem|A theorem]] of [[Charles Hermite|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>{{Citation| last1=Cohen| first1=Henri| author-link=Henri Cohen (number theorist)| last2=Diaz y Diaz| first2=Francisco | last3=Olivier| first3=Michel| contribution=A Survey of Discriminant Counting| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002| editor-last=Fieker| editor-first=Claus| editor2-last=Kohel| editor2-first=David R.| publisher=Springer-Verlag| location=Berlin| series=Lecture Notes in Computer Science | issn=0302-9743| isbn=978-3-540-43863-2| doi=10.1007/3-540-45455-1_7| year=2002| volume=2369| pages=80–94| mr=2041075}}</ref>

Let ''K'' be an algebraic number field, and let ''O<sub>K</sub>'' be its [[ring of integers]]. Let ''b''<sub>1</sub>, ..., ''b<sub>n</sub>'' be an [[integral basis]] of ''O<sub>K</sub>'' (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ<sub>1</sub>, ..., σ<sub>''n''</sub>} be the set of embeddings of ''K'' into the [[Complex number|complex numbers]] (i.e. [[injective]] [[Ring homomorphism|ring homomorphisms]] ''K''&nbsp;→&nbsp;'''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ<sub>''i''</sub>(''b<sub>j</sub>''). 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 [[Field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|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''&nbsp;=&nbsp;'''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 form|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 (number)|1]], [[5 (number)|5]], [[8 (number)|8]], [[12 (number)|12]], [[13 (number)|13]], [[17 (number)|17]], [[21 (number)|21]], [[24 (number)|24]], [[28 (number)|28]], [[29 (number)|29]], [[33 (number)|33]] (sequence A003658 in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).

The first eleven negative fundamental discriminants are:

: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in the [[On-Line Encyclopedia of Integer Sequences|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 inverse]]s 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]].{{cn|date=August 2024}}

==References==
{{reflist|colwidth=30em}}

==External links==
*[http://mathworld.wolfram.com/PolynomialDiscriminant.html Wolfram Mathworld: Polynomial Discriminant]
*[http://planetmath.org/discriminant Planetmath: Discriminant]

{{Polynomials}}

[[Category:Polynomials]]
[[Category:Conic sections]]
[[Category:Quadratic forms]]
[[Category:Determinants]]
[[Category:Algebraic number theory]]