Geometrical properties of polynomial roots

In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n 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 n with real coefficients, which is less than ${\displaystyle 1+{\frac {2}{\pi }}\ln(n)}$ for n sufficiently large.

In this article, a polynomial that is considered is always denoted

${\displaystyle p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n},}$

where ${\displaystyle a_{0},\dots ,a_{n}}$ are real or complex numbers and ${\displaystyle a_{n}\neq 0}$; thus ${\displaystyle n}$ is the degree of the polynomial.