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 alsoCubic Equation,
Polynomial,
Quadratic Equation,
Quartic Equation,
Resultant,
Root Separation,
Subresultant,
Vieta's Formulas Explore with Wolfram|Alpha ReferencesAkritas, 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|AlphaPolynomial Discriminant Cite this as:Weisstein, Eric W. "Polynomial Discriminant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolynomialDiscriminant.html
Subject classificationsRetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4