Geometrical properties of polynomial roots
In mathematics, a univariate polynomial of degree with real or complex coefficients has complex roots, if counted with their multiplicities. They form a multiset of points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plane that can be deduced from the degree and the coefficients of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree with real coefficients, which is less than for sufficiently large. In this article, a polynomial that is considered is always denoted where are real or complex numbers and ; thus is the degree of the polynomial. Continuous dependence on coefficients The roots of a polynomial of degree depend continuously on the coefficients. For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial). A consequence is that, for classical numeric root-finding algorithms, the problem of approximating the roots given the coefficients is ill-conditioned for many inputs. Conjugation The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form . It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be interpreted geometrically. Bounds on all roots Upper bounds on the absolute values of polynomial roots are widely used for root-finding algorithms, either for limiting the regions where roots should be searched, or for the computation of the computational complexity of these algorithms. Many such bounds have been given, and the sharper one depends generally on the specific sequence of coefficient that are considered. Most bounds are greater or equal to one, and are thus not sharp for a polynomial which have only roots of absolute values lower than one. However, such polynomials are very rare, as shown below. Any upper bound on the absolute values of roots provides a corresponding lower bound. In fact, if and is an upper bound of the absolute values of the roots of then is a lower bound of the absolute values of the roots of since the roots of