Discriminant
55607
218352013
2008-06-10T07:15:49Z
193.147.165.54
/* Formula */
In [[algebra]], the '''discriminant''' of a [[polynomial]] with [[real number|real]] or [[complex number|complex]] [[coefficients]] is a certain expression in the coefficients of the polynomial which is equal to zero if and only if the polynomial has a multiple root (i.e. a root with [[multiplicity]] greater than one) in the [[complex numbers]]. For example, the discriminant of the quadratic polynomial
:<math>ax^2+bx+c\,</math> is <math>\,b^2-4ac</math>.
The discriminant of the cubic polynomial
:<math>ax^3+bx^2+cx+d\,</math> is <math>\,b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd</math>.
This concept also applies if the polynomial has coefficients in a [[field (mathematics)|field]] which is not contained in the complex numbers. In this case, the discriminant vanishes if and only if the polynomial has multiple roots in its [[splitting field]]. The discriminant is given by
:<math>a_n^{2n-2}\prod_{i<j}{(r_i-r_j)^2}</math>
where <math>a_n</math> is the leading coefficient and <math>r_1, ..., r_n</math> are the roots (counting [[multiplicity]]) of the polynomial in some splitting field.
The concept of discriminant has been generalized to other [[algebraic structures]] besides polynomials, including [[conic section]]s, [[quadratic form]]s, and [[Discriminant of an algebraic number field|algebraic number fields]]. Discriminants in [[algebraic number theory]] are closely related, and contain information about [[ramification]]. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.
==Formula==
*The quadratic polynomial <math>ax^2+bx+c</math> has discriminant
: <math>\Delta=b^2-4ac;\,</math>
*the cubic polynomial <math>ax^3+bx^2+cx+d</math> has discriminant
: <math>\Delta=b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.\,</math>
Simpler polynomials have simpler expressions for their discriminants. For example,
*the [[monic polynomial|monic]] quadratic polynomial <math>x^2+bx+c</math> has discriminant
: <math>\Delta=b^2-4c;\,</math>
*the monic cubic polynomial <math>x^3+bx^2+cx+d</math> has discriminant
: <math>\Delta=b^2c^2-4c^3-4b^3d-27d^2+18bcd;\,</math>
*the monic cubic polynomial without quadratic term <math>x^3+px+q</math> has discriminant
: <math>\Delta=-4p^3-27q^2.\,</math>
==Quadratic formula==
The [[quadratic polynomial]] ''P''(''x'') = ''ax''<sup>2</sup> + ''bx'' + ''c'' has discriminant ''D'' = ''b''<sup>2</sup> − 4''ac'', which is the quantity under the square root sign in the [[quadratic formula]]. For real numbers a, b, c, one has:
<ul><li>When ''D'' > 0 , ''P''(''x'') has two distinct real roots <math>x_{1,2}=\frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>, and its graph crosses the ''x''-axis twice.
<li>When ''D'' = 0, ''P''(''x'') has two coincident real roots <math>x_1=x_2=-\frac{b}{2a}</math>, and its graph is tangent to the ''x''-axis.
<li>When ''D'' < 0 , ''P''(''x'') has no real roots, and its graph lies strictly above or below the ''x''-axis.
</ul>
==Discriminant of a polynomial==
The discriminant of the general polynomial
:<math>p(x)=a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_1 x+a_0</math>
is, up to a factor, equal to the [[determinant]] of the (2''n'' − 1)×(2''n'' − 1) matrix (see [[Sylvester matrix]])
:<math>\left(\begin{matrix}
& a_n & a_{n-1} & a_{n-2} & \ldots & a_1 & a_0 & 0 \ldots & \ldots & 0 \\
& 0 & a_n & a_{n-1} & a_{n-2} & \ldots & a_1 & a_0 & 0 \ldots & 0 \\
& \vdots\ &&&&&&&&\vdots\\
& 0 & \ldots\ & 0 & a_n & a_{n-1} & a_{n-2} & \ldots & a_1 & a_0 \\
& na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \ldots\ & 1a_1 & 0 & \ldots &\ldots & 0 \\
& 0 & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \ldots\ & 1a_1 & 0 & \ldots & 0 \\
& \vdots\ &&&&&&&&\vdots\\
& 0 & 0 & \ldots & 0 & 0 & na_n & (n-1)a_{n-1} & \ldots\ & 1a_1 \\
\end{matrix}\right).</math>
The determinant of this matrix is known as the [[resultant]] of <math>p(x)</math> and <math>p'(x)</math>, notation <math>R(p,p')</math>. The discriminant <math>D(p)</math> of <math>p(x)</math> is now given by the formula
:<math>D(p)=(-1)^{\frac{1}{2}n(n-1)}\frac{1}{a_n}R(p,p')\,</math>.
For example, in the case ''n'' = 4, the above determinant is
:<math>\begin{vmatrix}
& a_4 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \\
& 0 & a_4 & a_3 & a_2 & a_1 & a_0 & 0 \\
& 0 & 0 & a_4 & a_3 & a_2 & a_1 & a_0 \\
& 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \\
& 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \\
& 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1& 0 \\
& 0 & 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 \\
\end{vmatrix}</math>
The discriminant of the degree 4 polynomial is then obtained from this determinant upon dividing by <math>a_4</math>.
Equivalently, the discriminant is equal to
:<math>a_n^{2n-2}\prod_{i<j}{(r_i-r_j)^2}</math>
where ''r''<sub>1</sub>, ..., ''r''<sub>''n''</sub> are the [[complex number|complex]] roots (counting [[multiplicity]]) of the polynomial ''p(x)'':
:<math>\begin{matrix}p(x)&=&a_n x^n+a_{n-1}x^{n-1}+\ldots+a_1 x+a_0\\
&=&a_n(x-r_1)(x-r_2)\ldots (x-r_n)\end{matrix}</math>
This second expression makes it clear that, ''p'' has a multiple root [[if and only if]] the discriminant is zero. (This multiple root can be complex.)
The discriminant can be defined for polynomials over arbitrary [[field (mathematics)|field]]s, in exactly the same fashion as above. The product formula involving the roots ''r''<sub>''i''</sub> remains valid; the roots have to be taken in some [[splitting field]] of the polynomial.
==Discriminant of a conic section==
For a [[conic section]] defined by the real polynomial:
:''ax''<sup>2</sup> + ''bxy'' + ''cy''<sup>2</sup> + ''dx'' + ''ey'' + ''f''= 0
the discriminant is equal to
:''b''<sup>2</sup> − 4''ac'',
and determines the [[shape]] of the conic section. If the discriminant is less than 0, the equation is of an [[ellipse]] or a [[circle]]. If the discriminant equals 0, the equation is that of a [[parabola]]. If the discriminant is greater than 0, the equation is that of a [[hyperbola]]. This formula will not work for degenerate cases (when the polynomial factors).
==Discriminant of a quadratic form==
There is a substantive generalization to [[quadratic form]]s ''Q'' over any [[field (mathematics)|field]] ''K'' of [[characteristic (algebra)|characteristic]] ≠ 2. These can be written as a sum of terms
:''a''<sub>''i''</sub>''L''<sub>''i''</sub><sup>2</sup>
where the ''L''<sub>''i''</sub> are linear forms and 1 ≤ ''i'' ≤ ''n'' where ''n'' is the number of variables. Then the '''discriminant''' is the product of the ''a''<sub>''i''</sub>, taken in ''K''/''K''<sup>2</sup>, and is then well defined (i.e., up to squares). A more invariant way to say this is as (the class of) the determinant of a [[symmetric matrix]] for ''Q''.
==Discriminant of an algebraic number field==
See main article, [[Discriminant of an algebraic number field]].
==External links==
*[http://mathworld.wolfram.com/PolynomialDiscriminant.html Mathworld article]
*[http://planetmath.org/encyclopedia/Discriminant.html Planetmath article]
[[Category:Polynomials]]
[[Category:conic sections]]
[[Category:Quadratic forms]]
[[Category:Determinants]]
[[Category:algebraic number theory]]
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