A polynomial discriminant is the product of the squares of the differences of the polynomial roots .
The discriminant of a polynomial is defined only up to constant factor, and several
slightly different normalizations can be used. For a polynomial
(1)
of degree ,
the most common definition of the discriminant is
(2)
which gives a homogenous polynomial of degree in the coefficients of .
The discriminant of a polynomial is given in terms of a resultant
as
(3)
where
is the derivative of and is the degree of . For fields of infinite characteristic, so the formula reduces to
(4)
The discriminant of a univariate polynomial is implemented in the Wolfram
Language as Discriminant [p ,
x ].
The discriminant of the quadratic equation
(5)
is given by
(6)
The discriminant of the cubic equation
(7)
is given by
(8)
The discriminant of a quartic equation
(9)
is
(10)
(Schroeppel 1972).
See also Cubic Equation ,
Polynomial ,
Quadratic Equation ,
Quartic
Equation ,
Resultant ,
Root
Separation ,
Subresultant ,
Vieta's
Formulas
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References Akritas, A. G. Elements of Computer Algebra with Applications. New York: Wiley, 1989. Basu,
S.; Pollack, R.; and Roy, M.-F. Algorithms
in Real Algebraic Geometry. Berlin: Springer-Verlag, 2003. Caviness,
B. F. and Johnson, J. R. (Eds.). Quantifier
Elimination and Cylindrical Algebraic Decomposition. New York: Springer-Verlag,
1998. Cohen, H. "Resultants and Discriminants." §3.3.2
in A
Course in Computational Algebraic Number Theory. New York: Springer-Verlag,
pp. 119-123, 1993. Cox, D.; Little, J.; and O'Shea, D. Ideals,
Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative
Algebra, 2nd ed. New York: Springer-Verlag, 1996. Mignotte, M.
and Stefănescu, D. Polynomials:
An Algorithmic Approach. Singapore: Springer-Verlag, 1999. Schroeppel,
R. Item 4 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge,
MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 4, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/geometry.html#item4 . Zippel,
R. Effective
Polynomial Computation. Boston, MA: Kluwer, 1993. Referenced on
Wolfram|Alpha Polynomial Discriminant
Cite this as:
Weisstein, Eric W. "Polynomial Discriminant."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/PolynomialDiscriminant.html
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