Editing Fundamental theorem of algebra (section)

==Bounds on the zeros of a polynomial==
{{main|Properties of polynomial roots}}
While the fundamental theorem of algebra states a general existence result, it is of some interest, both from the theoretical and from the practical point of view, to have information on the location of the zeros of a given polynomial. The simplest result in this direction is a bound on the modulus: all zeros ζ of a monic polynomial <math>z^n+a_{n-1}z^{n-1}+\cdots+a_1z +a_0</math> satisfy an inequality |ζ| ≤ ''R''<sub>∞</sub>, where

:<math>R_{\infty}:= 1+\max\{|a_0|,\ldots,|a_{n-1}|\}. </math>

As stated, this is not yet an existence result but rather an example of what is called an [[a priori and a posteriori|a priori]] bound: it says that ''if there are solutions'' then they lie inside the closed disk of center the origin and radius ''R''<sub>∞</sub>. However, once coupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. More generally, a bound can be given directly in terms of any [[p-norm]] of the ''n''-vector of coefficients <math>a:=( a_0, a_1, \ldots, a_{n-1}),</math> that is |ζ| ≤ ''R<sub>p</sub>'', where ''R<sub>p</sub>'' is precisely the ''q''-norm of the 2-vector <math>(1, \|a\|_p),</math> ''q'' being the conjugate exponent of ''p'', <math>\tfrac{1}{p} + \tfrac{1}{q} =1,</math> for any 1 ≤ ''p'' ≤ ∞. Thus, the modulus of any solution is also bounded by

:<math> R_1:= \max\left \{ 1 , \sum_{0\leq k<n} |a_k|\right \},</math>
:<math> R_p:= \left[ 1 + \left(\sum_{0\leq k<n}|a_k|^p\right )^{\frac{q}{p}}\right ]^{\frac{1}{q}},</math>

for 1 < ''p'' < ∞, and in particular

:<math> R_2:= \sqrt{\sum_{0\leq k\leq n} |a_k|^2 }</math>

(where we define ''a<sub>n</sub>'' to mean 1, which is reasonable since 1 is indeed the ''n''-th coefficient of our polynomial). The case of a generic polynomial of degree ''n'',

:<math>P(z):= a_n z^n+a_{n-1}z^{n-1}+\cdots+a_1z +a_0,</math>

is of course reduced to the case of a monic, dividing all coefficients by ''a<sub>n</sub>'' ≠ 0. Also, in case that 0 is not a root, i.e. ''a''<sub>0</sub> ≠ 0, bounds from below on the roots ζ follow immediately as bounds from above on <math>\tfrac{1}{\zeta}</math>, that is, the roots of

:<math>a_0 z^n+a_1z^{n-1}+\cdots+a_{n-1}z +a_n.</math>

Finally, the distance <math>|\zeta-\zeta_0|</math> from the roots ζ to any point <math>\zeta_0</math> can be estimated from below and above, seeing <math>\zeta-\zeta_0</math> as zeros of the polynomial <math>P(z+\zeta_0)</math>, whose coefficients are the [[Taylor expansion]] of ''P''(''z'') at <math>z=\zeta_0.</math>

Let ζ be a root of the polynomial

:<math>z^n+a_{n-1}z^{n-1}+\cdots+a_1z +a_0;</math>

in order to prove the inequality |ζ| ≤ ''R<sub>p</sub>'' we can assume, of course, |ζ| > 1. Writing the equation as

:<math>-\zeta^n=a_{n-1}\zeta^{n-1}+\cdots+a_1\zeta+a_0,</math>

and using the [[Hölder's inequality]] we find

:<math>|\zeta|^n\leq \|a\|_p \left \| \left (\zeta^{n-1},\ldots,\zeta, 1 \right ) \right \|_q.</math>

Now, if ''p'' = 1, this is

:<math>|\zeta|^n\leq\|a\|_1\max \left \{|\zeta|^{n-1},\ldots,|\zeta|,1 \right \} =\|a\|_1|\zeta|^{n-1},</math>

thus

:<math>|\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>|\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>|\zeta|^{nq}\leq \|a\|_p^q \frac{|\zeta|^{qn}}{|\zeta|^q-1}</math>

and simplifying,

:<math>|\zeta|^q\leq 1+\|a\|_p^q.</math>

Therefore

:<math>|\zeta|\leq \left \| \left (1,\|a\|_p \right ) \right \|_q=R_p </math>

holds, for all 1 ≤ ''p'' ≤ ∞.