Properties of polynomial roots
From Wikipedia, the free encyclopedia
In mathematics, a polynomial is a function of the form
where the coefficients 
are complex numbers. The fundamental theorem of algebra states that polynomial p has n roots. The aim of this page is to list various properties of these roots.
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[edit] Continuous dependence of coefficients
The n roots of an polynomial of degree n depend continuously on the coefficients. This means that there are n continuous functions 
depending on the coefficients that parametrize the roots with correct multiplicity.
This result implies that the eigenvalues of a matrix depend continuously on the matrix. A proof can be found in Tyrtyshnikov(1997).
The problem of approximating the roots given the coefficients is ill-conditioned. See, for example, Wilkinson's polynomial.
[edit] Complex conjugate root theorem
The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the roots appear in pairs of the type 
.
For example, the equation x2 + 1 = 0 has roots 
.
[edit] Gauss-Lucas theorem
The Gauss-Lucas theorem states that the convex hull of the roots of a polynomial contains the roots of the derivative of the polynomial.
A sometimes useful corollary is that if all roots of a polynomial have positive real part, then so do the roots of all derivatives of the polynomial.
A related result is the Bernstein's inequality. It states that for a polynomial P of degree n with derivative P′ we have
[edit] See also
[edit] References
- E.E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis, Birkhäuser Boston, 1997
