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Polynomial Discriminant


A polynomial discriminant is the product of the squares of the differences of the polynomial roots r_i. The discriminant of a polynomial is defined only up to constant factor, and several slightly different normalizations can be used. For a polynomial

 p(z)=a_nz^n+a_(n-1)z^(n-1)+...+a_1z+a_0
(1)

of degree n, the most common definition of the discriminant is

 D(p)=a_n^(2n-2)product_(i,j; i<j)^n(r_i-r_j)^2,
(2)

which gives a homogenous polynomial of degree 2(n-1) in the coefficients of p.

The discriminant of a polynomial p is given in terms of a resultant as

 D(p)=(-1)^(n(n-1)/2)R(p,p^')a_n^(n-k-2),
(3)

where p^' is the derivative of p and k is the degree of p^'. For fields of infinite characteristic, k=n-1 so the formula reduces to

 D(p)=((-1)^(n(n-1)/2)R(p,p^'))/(a_n).
(4)

The discriminant of a univariate polynomial p(x) is implemented in the Wolfram Language as Discriminant[p, x].

The discriminant of the quadratic equation

 a_2z^2+a_1z+a_0=0
(5)

is given by

 D_2=a_1^2-4a_0a_2.
(6)

The discriminant of the cubic equation

 a_3z^3+a_2z^2+a_1z+a_0=0
(7)

is given by

 D_3=a_1^2a_2^2-4a_0a_2^3-4a_1^3a_3+18a_0a_1a_2a_3-27a_0^2a_3^2
(8)

The discriminant of a quartic equation

 a_4z^4+a_3z^3+a_2z^2+a_1z+a_0=0
(9)

is

 D_4=[(a_1^2a_2^2a_3^2-4a_1^3a_3^3-4a_1^2a_2^3a_4+18a_1^3a_2a_3a_4-27a_1^4a_4^2+256a_0^3a_4^3)+a_0(-4a_2^3a_3^2+18a_1a_2a_3^3+16a_2^4a_4-80a_1a_2^2a_3a_4-6a_1^2a_3^2a_4+144a_1^2a_2a_4^2)+a_0^2(-27a_3^4+144a_2a_3^2a_4-128a_2^2a_4^2-192a_1a_3a_4^2)]
(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.

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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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