1. Continuous Dependence on Coefficients

The n roots of a polynomial of degree n depend continuously on the coefficients.

This result implies that the eigenvalues of a matrix depend continuously on the matrix. A proof can be found in a book of Tyrtyshnikov.^[1]

The problem of approximating the roots given the coefficients is ill-conditioned. See, for example, Wilkinson's polynomial.

2. Algebraic Expression of Roots

All roots of polynomials with rational coefficients are algebraic numbers, by definition of the latter. If the degree n of the polynomial is no greater than 4, all the roots of the polynomial can be written as algebraic expressions in terms of the coefficients—that is, applying only the four basic arithmetic operations and the extraction of n-th roots. By the Abel–Ruffini theorem, this cannot be done in general for higher-degree equations.

3. Complex Conjugate Root Theorem

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 type a ± ib.

For example, the equation x² + 1 = 0 has roots ± i.

4. Radical Conjugate Roots

If a polynomial P(x) with rational coefficients has a + √b as a root, where a, b are rational and √b is irrational, then a − √b is also a root. First observe that

[math]\displaystyle{ \left(x - \left [ a + \sqrt b \right ] \right) \left(x - \left [ a - \sqrt b \right ] \right) = (x - a)^2 - b. }[/math]

Denote this quadratic polynomial by D(x). Then, by the Euclidean division of polynomials,

[math]\displaystyle{ P(x) = D(x)Q(x) + cx + d = ((x - a)^2 - b)Q(x) + cx + d, \,\! }[/math]

where c, d are rational numbers (by virtue of the fact that the coefficients of P(x) and D(x) are all rational). But a + √b is a root of P(x):

[math]\displaystyle{ P\left( a + \sqrt b \right) = c\left(a + \sqrt b \right) + d = (ac + d) + c \sqrt b = 0. }[/math]

It follows that c, d must be zero, since otherwise the final equality could be arranged to suggest the irrationality of rational values (and vice versa). Hence P(x) = D(x)Q(x), for some quotient polynomial Q(x), and D(x) is a factor of P(x).^[2]

This property may be generalized, with the same proof, as: If a + √b is a root of the polynomial P(x), and √b does not belong to the smallest field containing a, b and the coefficients of P(x), then a − √b is also a root of P(x).

The property may be further generalized into: If an irreducible polynomial P has a root in common with a polynomial Q, then all roots of P are roots of Q, and P divides Q evenly.

5. Bounds on (Complex) Polynomial Roots

5.1. Based on the Rouché Theorem

A very general class of bounds on the magnitude of roots is implied by the Rouché theorem. If there is a positive real number R and a coefficient index k such that

[math]\displaystyle{ |a_k|\,R^k \gt |a_0|+\cdots+|a_{k-1}|\,R^{k-1}+|a_{k+1}|\,R^{k+1}+\cdots+|a_n|\,R^n }[/math]

then there are exactly k (counted with multiplicity) roots of absolute value less than R. For k=0,n there is always a solution to this inequality, for example

• for k=n,

[math]\displaystyle{ R=1+\frac1{|a_n|}\max\{|a_0|,|a_1|,\dots, |a_{n-1}|\} }[/math] or

[math]\displaystyle{ R=\max\left(1,\,\frac1{|a_n|}\left(|a_0|+|a_1|+\cdots+|a_{n-1}|\right)\right) }[/math]

are upper bounds for the size of all roots,

• for k=0,

[math]\displaystyle{ R=\frac{|a_0|}{|a_0|+\max\{|a_1|,|a_2|,\dots, |a_{n}|\}} }[/math] or

[math]\displaystyle{ R=\frac{|a_0|}{\max(|a_0|,\,|a_1|+|a_2|+\cdots+|a_{n}|)} }[/math]

are lower bounds for the size of all of the roots.

• for all other indices, the function

[math]\displaystyle{ h(R)=|a_0|\,R^{-k}+\cdots+|a_{k-1}|\,R^{-1}-|a_k|+|a_{k+1}|\,R+\cdots+|a_n|\,R^{n-k} }[/math]

is convex on the positive real numbers, thus the minimizing point is easy to determine numerically. If the minimal value is negative, one has found additional information on the location of the roots.

One can increase the separation of the roots and thus the ability to find additional separating circles from the coefficients, by applying the root squaring operation of the Dandelin-Graeffe iteration to the polynomial.

A different approach is by using the Gershgorin circle theorem applied to some companion matrix of the polynomial, as it is used in the Weierstraß–(Durand–Kerner) method. From initial estimates of the roots, that might be quite random, one gets unions of circles that contain the roots of the polynomial.

5.2. Other Bounds

Useful upper bounds for the magnitudes of all of a polynomial's roots^[3] include the near optimal Fujiwara bound^[4]

[math]\displaystyle{ 2\, \max \left\{ \left|\frac{a_{n-1}}{a_n}\right|, \left|\frac{a_{n-2}}{a_n}\right|^{\frac{1}{2}}, \cdots, \left|\frac{a_1}{a_n}\right|^\frac{1}{n-1}, \left|\frac{a_0}{2a_n}\right|^\frac 1 n\right\}, }[/math]

which is an improvement (as the geometric mean) of Kojima's bound:^[5]

[math]\displaystyle{ 2\,\max \left\{ \left|\frac{a_{n-1}}{a_n}\right|,\left|\frac{a_{n-2}}{a_{n-1}}\right|, \cdots, \left|\frac{a_0}{2a_1}\right|\right\}. }[/math]

Other bounds are the Cauchy bound^[6]

[math]\displaystyle{ 1+\max\left\{ \left|\frac{a_{n-1}}{a_n}\right|, \left|\frac{a_{n-2}}{a_{n}}\right|, \cdots, \left|\frac{a_0}{a_n}\right|\right\} }[/math]

and the Lagrange bound^[7][8]

[math]\displaystyle{ \max\left\{1,\sum_{i=0}^{n-1} \left|\frac{a_i}{a_n}\right|\right\} }[/math]

or

[math]\displaystyle{ \sum_{i=0}^{n-1} \left|\frac{a_i}{a_{i+1}}\right|. }[/math]

These expressions return only bounds surpassing unity, so they cannot be used for some polynomials.^{[clarification needed]}

Without loss of generality we assume the polynomial to be monic with general term a_ixⁱ. Let { a_j } be the set of negative coefficients. An upper bound for the positive real roots is given by the sum of the two largest numbers in the set { |a_j|^1/j }. This is an improvement on Fujiwara's bound which uses twice the maximum value of this set as its upper bound.

A similar bound also due to Lagrange holds for a polynomial with complex coefficients. Again assume the polynomial to be monic with general term a_ixⁱ. Then the upper bound for the absolute values of the roots is given by the sum of the two greatest values in the set { |a_i|^1/i }. Again this is an improvement on Fujiwara's bound which uses twice the maximum value of this set as its upper bound.

A third bound also due to Lagrange holds for a polynomial with real coefficients. Let the a_ix^n-i be the general term of the polynomial with 0 ≤ i ≤ m. Let the first d terms of the polynomial have positive coefficients and let A be the maximum of these d coefficients. Then 1 + ( A / a₀ )^{1/( 1 + d )} is an upper bound to the positive roots of the polynomial.

Sun and Hsieh obtained an improvement on Cauchy's bound.^[9] Assume the polynomial is monic with general term a_ixⁱ. Sun and Hsieh showed that upper bounds 1 + d₁ and 1 + d₂ could be obtained from the following equations.

[math]\displaystyle{ d_1 = \tfrac{1}{2} \left((| a_{n-1}| - 1) + \sqrt{(|a_{n-1}| - 1 )^2 + 4a } \right), \qquad a = \max \{ |a_i | \}. }[/math]

d₂ is the positive root of the cubic equation

[math]\displaystyle{ Q(x) = x^3 + (2 - |a_{n-1}|) x^2 + (1 - |a_{n-1}| - |a_{n-2}| ) x - a, \qquad a = \max \{ |a_i | \} }[/math]

They also noted that d₂ ≤ d₁

Proof

Let ζ be a root of the polynomial

[math]\displaystyle{ z^n+a_{n-1}z^{n-1}+\cdots+a_1z +a_0; }[/math]

in order to prove the inequality |ζ| ≤ R_p we can assume, of course, |ζ| > 1. Writing the equation as

[math]\displaystyle{ -\zeta^n=a_{n-1}\zeta^{n-1}+\cdots+a_1\zeta+a_0, }[/math]

and using the Hölder's inequality we find

[math]\displaystyle{ |\zeta|^n\leq \|a\|_p \left \|(\zeta^{n-1},\cdots,\zeta, 1) \right \|_q. }[/math]

Now, if p = 1, this is

[math]\displaystyle{ |\zeta|^n\leq\|a\|_1\max \left \{|\zeta|^{n-1},\cdots,|\zeta|,1 \right \} =\|a\|_1|\zeta|^{n-1}, }[/math]

thus

[math]\displaystyle{ |\zeta|\leq \max\{1,\|a\|_1\}. }[/math]

In the case 1 < p ≤ ∞, taking into account the summation formula for a geometric progression, we have

[math]\displaystyle{ |\zeta|^n\leq \|a\|_p \left(|\zeta|^{q(n-1)}+\cdots+|\zeta|^q +1\right)^{\frac{1}{q}}=\|a\|_p \left(\frac{|\zeta|^{qn}-1}{|\zeta|^q-1}\right)^{\frac{1}{q}}\leq\|a\|_p \left(\frac{|\zeta|^{qn}}{|\zeta|^q-1}\right)^{\frac{1}{q}}, }[/math]

thus

[math]\displaystyle{ |\zeta|^{nq}\leq \|a\|_p^q \frac{|\zeta|^{qn}}{|\zeta|^q-1} }[/math]

simplifying we get

[math]\displaystyle{ |\zeta|^q\leq 1+\|a\|_p^q. }[/math]

Therefore

[math]\displaystyle{ |\zeta|\leq \left \| \left (1,\|a\|_p \right ) \right \|_q=R_p, }[/math]

holds, for all 1 ≤ p ≤ ∞.

5.3. Landau's Inequality

The previous bounds are upper bounds for each root separately. Landau's inequality provides an upper bound for the absolute values of the product of the roots that have an absolute value greater than one. This bound for the product of roots is not much greater than the preceding bounds of each root separately.^[10]

Let [math]\displaystyle{ z_1, \ldots, z_n }[/math] be the n roots of the polynomial p, and

[math]\displaystyle{ M(p)=|a_n|\prod_{j=1}^n \max(1,|z_j|). }[/math]

Then

[math]\displaystyle{ M(p)\le \sqrt{\sum_{k=0}^n |a_k|^2}\,. }[/math]

This bound is useful to bound the coefficients of a divisor of a polynomial: if

[math]\displaystyle{ q= \sum_{k=0}^m b_k x^k }[/math]

is a divisor of p, then

[math]\displaystyle{ \sum_{k=0}^m |b_k| \le 2^m\,\left | \frac{b_m}{a_n}\right |\, M(p)\,. }[/math]

5.4. Bounds on Positive Polynomial Roots

There also exist bounds on just the positive roots of polynomials; these bounds were developed by Akritas, Strzeboński and Vigklas based on previous work by Doru Stefanescu. They are used in the computer algebra systems Mathematica, SageMath, SymPy, Xcas etc.^[11][12]

5.5. Polynomials with Real Roots

It is possible to determine the bounds of the roots of a polynomial using Samuelson's inequality. This method is due to a paper by Laguerre.^[13]

Let [math]\displaystyle{ \sum_{k=0}^n a_k x^k }[/math] be a polynomial with all real roots. Then its roots are located in the interval with endpoints

[math]\displaystyle{ x_\pm=-\frac{a_{n-1}}{na_n} \pm \frac{n-1}{na_n}\sqrt{a^2_{n-1} - \frac{2n}{n-1}a_n a_{n-2}}. }[/math]

Example: The polynomial [math]\displaystyle{ x^4+5x^3+5x^2-5x-6 }[/math] has four real roots −3, −2, −1 and 1. The above formula gives

[math]\displaystyle{ x_\pm=-\frac{5}{4} \pm \frac{3}{4}\sqrt{\frac{35}{3}}; }[/math]

thus its roots are contained in [math]\displaystyle{ I = [-3.8117, 1.3117] }[/math].

6. 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 Bernstein's inequality. It states that for a polynomial P of degree n with derivative P′ we have

[math]\displaystyle{ \max_{|z| \leq 1} \big|P'(z)\big| \le n \max_{|z| \leq 1} \big|P(z)\big|. }[/math]

7. Statistical Distribution of the Roots

The statistical properties of the roots of a random polynomial have been the subject of several studies. Let

[math]\displaystyle{ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 }[/math]

be a random polynomial. If the coefficients a_i are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be on or close to the unit circle.

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^[14][15]

[math]\displaystyle{ m( x ) = \frac { \sqrt{ A( x ) C( x ) - B( x )^2 }} {\pi A( x )} }[/math]

where

[math]\displaystyle{ \begin{align} A( x ) &= \sigma \sum { x^{ 2i } } = \sigma \frac{ x^{ 2n } - 1 } { x - 1 }, \\ B( x ) &= \frac{ 1 } { 2 } \frac{ d } { dx } A( x ), \\ C( x ) &= \frac{ 1 } { 4 } \frac{ d^2 } { dx^2 } A( x ) + \frac{ 1 } { 4x } \frac{ d } { dx } A( x ). \end{align} }[/math]

When the coefficients are Gaussian distributed with a non-zero mean and variance of σ, a similar but more complex formula is known.

7.1. Asymptotic Results

For large n, a number of asymptotic formulae are known. For a fixed x

[math]\displaystyle{ m( x ) = \frac{ 1 } { \pi | 1 - x^2 | } }[/math]

and

[math]\displaystyle{ m( \pm 1 ) = \frac{ 1 } { \pi } \sqrt { \frac{ n^2 - 1 } { 12 } } }[/math]

where m( x ) is the mean density of real roots. The expected number of real roots is

[math]\displaystyle{ N_n = \frac{ 2 } { \pi } \ln n + C + O( n^{ -2 } ) }[/math]

where C is a constant approximately equal to 0.6257358072 and O() is the order operator.

This result has been shown by Kac, Erdös and others to be insensitive to the actual distribution of coefficients. Numerical testing of this formula has confirmed these earlier results.