Fundamental Theorem of Algebra - Bounds On The Zeroes of A Polynomial

Bounds On The Zeroes of A Polynomial

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 zeroes of a given polynomial. The simpler result in this direction is a bound on the modulus: all zeroes of a monic polynomial satisfy an inequality where

Notice that, as stated, this is not yet an existence result but rather an example of what is called an a priori bound: it says that if there are solutions then they lie inside the closed disk of center the origin and radius . 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, that is, where is precisely the q-norm of the 2-vector, q being the conjugate exponent of p, 1/p + 1/q = 1, for any . Thus, the modulus of any solution is also bounded by

for, and in particular

(where we define 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, is of course reduced to the case of a monic, dividing all coefficients by . Also, in case that 0 is not a root, i.e., bounds from below on the roots follow immediately as bounds from above on, that is, the roots of . Finally, the distance from the roots to any point can be estimated from below and above, seeing as zeroes of the polynomial, whose coefficients are the Taylor expansion of at

We report here the proof of the above bounds, which is short and elementary. Let be a root of the polynomial ; in order to prove the inequality we can assume, of course, . Writing the equation as, and using the Hölder's inequality we find . Now, if, this is, thus . In the case, taking into account the summation formula for a geometric progression, we have

thus and simplifying, . Therefore holds, for all