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.
For simple roots, this results immediately from the implicit function theorem.
A consequence is that, for classical numeric root-finding algorithms, the problem of approximating the roots given the coefficients can be ill-conditioned for many inputs.
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.
Lagrange and Cauchy were the first to provide upper bounds on all complex roots.
This is relatively rare in practice, and explains why Cauchy's bound is more widely used than Lagrange's.
Both bounds result from the Gershgorin circle theorem applied to the companion matrix of the polynomial and its transpose.
denotes the ith nonzero coefficient when the terms of the polynomials are sorted by increasing degrees.
Hölder's inequality allows the extension of Lagrange's and Cauchy's bounds to every h-norm.
denotes the ith nonzero coefficient when the terms of the polynomials are sorted by increasing degrees.
Sun and Hsieh showed that upper bounds 1 + d1 and 1 + d2 could be obtained from the following equations.
This inequality, discovered in 1905 by Edmund Landau,[9] has been forgotten and rediscovered at least three times during the 20th century.
Thus and Rouché's theorem allows defining discs centered at zero and containing a given number of roots.
More precisely, if there is a positive real number R and an integer 0 ≤ k ≤ n such that then there are exactly k roots, counted with multiplicity, of absolute value less than R.
For k = 0 and k = n, Descartes' rule of signs shows that the polynomial has exactly one positive real root.
these bounds are optimal for polynomials with a given sequence of the absolute values of their coefficients.
If the roots have distinct absolute values, one can eventually completely separate the roots in terms of their absolute values, that is, compute n + 1 positive numbers
for k = 1, ..., n. The Gershgorin circle theorem applies the companion matrix of the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing a root of the polynomial; see Durand–Kerner method § Root inclusion via Gerschgorin's circles for details.
For example, the efficiency of the method of continued fractions for real-root isolation strongly depends on tightness of a bound of positive roots.
If this bound is not greater than 1, this means that all nonzero coefficients have the same sign, and that there is no positive root.
Other bounds have been recently developed, mainly for the method of continued fractions for real-root isolation.
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and essentially the same absolute values of the coefficient.
It states that for a polynomial P of degree n with derivative P′ we have If the coefficients ai of a random polynomial are independently and identically distributed with a mean of zero, most complex roots are on the unit circle or close to it.
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula[21][22] where When the coefficients are Gaussian distributed with a non-zero mean and variance of σ, a similar but more complex formula is known.
[citation needed] For large n, the mean density of real roots near x is asymptotically if
and It follows that the expected number of real roots is, using big O notation where C is a constant approximately equal to 0.6257358072.
As a result, most root-finding algorithms suffer substantial loss of accuracy on multiple roots in numerical computation.
In 1972, William Kahan proved that there is an inherent stability of multiple roots.