This is related to the observation that, in the purely real realm, if you take some positive real number, and repeatedly take its roots (e.g just keep hitting the square root button on your calculator), you will reach a fixed point of 1.0. Gee, why is the n-th root of any positive real number, even a large one, so close to 1.0?
It's because you can reconstruct that large number by taking a sufficiently large power of that 1.0 + [small delta].
(Same thing with a small one: start with some .000000000000xxx and take roots: it blows up quickly toward 1.0).
It's the same with the polynomial. The n roots are just numbers that let us write the polynomial as:
P(z) = (z - r0)(z - r1)...(z - rn-1)
Their modulus |ri| doesn't have to be far from 1 for them be able to reproduce P(z), even if if some of the coefficients are decepively large when it's written as the power series.
If one of the ri's is far out somewhere, then the (z - ri) term will contribute a wild factor. Several of these will just blow the thing way out.
It's because you can reconstruct that large number by taking a sufficiently large power of that 1.0 + [small delta].
(Same thing with a small one: start with some .000000000000xxx and take roots: it blows up quickly toward 1.0).
It's the same with the polynomial. The n roots are just numbers that let us write the polynomial as:
P(z) = (z - r0)(z - r1)...(z - rn-1)
Their modulus |ri| doesn't have to be far from 1 for them be able to reproduce P(z), even if if some of the coefficients are decepively large when it's written as the power series.
If one of the ri's is far out somewhere, then the (z - ri) term will contribute a wild factor. Several of these will just blow the thing way out.