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单词 Discriminant
释义

discriminant


Summary.

The discriminantMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a given polynomialMathworldPlanetmathPlanetmathPlanetmath is a number, calculatedfrom the coefficients of that polynomial, that vanishes if and only ifthat polynomial has one or more multiple roots. Using thediscriminant we can test for the presence of multiple roots, withouthaving to actually calculate the roots of the polynomial in question.

There are other ways to do this of course; one can look at the formal derivative of the polynomial (it will be coprimeMathworldPlanetmathPlanetmath to the original polynomial if and only if that original had no multiple roots). But the discriminant turns out to be valuable in a number of other contexts. For example, we will see that the discriminant of X2+bX+c is b2-4c; the quadratic formula states that the roots are -b/2±b2-4c/2, so that the discriminant also determines whether the roots of this polynomial are real or not. In higher degrees, its role is more complicated.

There are other uses of the word “discriminant” that are closely related to this one.If (α) is a number fieldMathworldPlanetmath, then the http://planetmath.org/node/2895discriminant of (α) is the discriminant of the minimal polynomialPlanetmathPlanetmath of α. For more general extensions of number fields, one must use a different definition of discriminant generalizing this one. If we have an elliptic curveMathworldPlanetmath over the rational numbers defined by the equation y2=x3+Ax+B, then its modular discriminant is the discriminant of the cubic polynomial on the right-hand side. For more on both these facts, see [1] on number fields and [2] on elliptic curves.

Definition.

The discriminant of order n is the polynomial, denotedhere 11 The discriminant of a polynomial p is oftentimesalso denoted as “disc(p)by δ(n)=δ(n)(a1,,an), characterized by the followingrelation:

δ(n)(s1,s2,,sn)=i=1nj=i+1n(xi-xj)2,(1)

where

sk=sk(x1,,xn),k=1,,n

is thekth elementary symmetric polynomial.

The above relation is a defining one, because the right-hand side of(1) is, evidently, a symmetric polynomialMathworldPlanetmath, and because the algebraMathworldPlanetmathPlanetmath ofsymmetric polynomials is freely generated by the basic symmetricpolynomials, i.e. every symmetric polynomial arises in a uniquefashion as a polynomial of s1,,sn.

Proposition 1.

The discriminant d of a polynomial may be expressed with the resultantMathworldPlanetmath R of the polynomial and its first derivativeMathworldPlanetmath:

d=(-1)n(n-1)2R/an

Proposition 2.

Up to sign, the discriminant is given by the determinantMathworldPlanetmath of a2n-1 square matrixMathworldPlanetmath with columns 1 to n-1 formed by shifting thesequence  1,a1,,an,  and columns n to 2n-1 formed byshifting the sequence  n,(n-1)a1,,an-1,  i.e.

δ(n)=|100n000a110(n-1)a1n00an-2an-312an-23an-3n0an-1an-2a1an-12an-2(n-1)a1nanan-1a20an-1(n-2)a2(n-1)a100an-100an-12an-200an000an-1|(2)

Multiple root test.

Let 𝕂 be a field, let x denote anindeterminate, and let

p=xn+a1xn-1++an-1x+an,ai𝕂

be a monic polynomialMathworldPlanetmath over 𝕂. We defineδ[p], the discriminant of p, by setting

δ[p]=δ(n)(a1,,an).

The discriminant of a non-monic polynomial is defined homogenizing theabove definition, i.e by setting

δ[ap]=a2n-2δ[p],a𝕂.
Proposition 3.

The discriminant vanishes if and only if p has multiple roots inits splitting fieldMathworldPlanetmath.

Proof.

It isn’t hard to show that a polynomial has multiple roots if and onlyif that polynomial and its derivative share a common root. Thedesired conclusion now follows by observing thatthe determinant formula in equation (2) gives theresolvent of a polynomial and its derivative. This resolvent vanishesif and only if the polynomial in question has a multiple root.∎

Some Examples.

Here are the first few discriminants.

δ(1)=1
δ(2)=a12-4a2
δ(3)=18a1a2a3+a12a22-4a23-4a13a3-27a32
δ(4)=a12a22a32-4a23a32-4a13a33+18a1a2a33-27a34
-4a12a23a4+16a24a4+18a13a2a3a4-80a1a22a3a4
-6a12a32a4+144a2a32a4-27a14a42+144a12a2a42
-128a22a42-192a1a3a42+256a43

Here is the matrix used to calculate δ(4):

δ(4)=|1004000a1103a1400a2a112a23a140a3a2a1a32a23a14a4a3a20a32a23a10a4a300a32a200a4000a3|

References

  • 1 Daniel A. Marcus, Number Fields, Springer, New York.
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

See also the bibliography for number theory (http://planetmath.org/BibliographyForNumberTheory) and the bibliography for algebraic geometry (http://planetmath.org/BibliographyForAlgebraicGeometry).

Titlediscriminant
Canonical nameDiscriminant
Date of creation2013-03-22 12:31:12
Last modified on2013-03-22 12:31:12
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id17
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 12E05
Synonympolynomial discriminant
Related topicResolvent
Related topicDiscriminantOfANumberField
Related topicModularDiscriminant
Related topicJInvariant
Related topicDiscriminantOfAlgebraicNumber
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