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, whose geometry 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.