Approximating roots

I have a polynomial to the N th degree, P(n), and need to approximate the roots, bound them. Then I know that x^n - x must be bound, so that the nth derivative is within some bound error of x. That gives the finite log value used up to the nth derivative. The solutions to the approximations are N equally space angles from -pi/2 to pi/2 along the hyperbolic curve, when the bound is one. All points where the sum of tanh(n) = pi, and n are equally spaced, that gives a geodesic approximation of the unit circle.
I can repeat the process recursively, as long as I subselect the range of valid x such that the next fractional errors are bound by the first fractional error. decrease as I apply the process repeatedly. Each time I break my error sphere into a sum of smaller error sphere having one less degree of freedom.

There is a simpler method.  Consider some r and its finite log approximation:

 Where we truncate the series. We want the approximation to go uniformly lower, so we limit ourselves to x such that: abs((x-1)^n)/n goes uniformly smaller as n goes from 1 to Nmax. so:
 x-1 < root(2), and (x-1) < root(3)... and they also go from large to small. Limiting the angle from 0 to pi, then I can construct the roots such that they are within some finite bound. The angle of any root for an nth degree polynomial is:

k* Pi/(n) + j*pi/(n-1) + i*pi/(n-2).... the coefficients limited to n-k+1.
The radius are all bounded by the previous error bounds. So:
 r(n-1)^2 = 1/2-(1/4+r(n))^2, I think this works, or some similar. This gets the real component of all the roots. The imaginary component defines the error function, the equations of motion. This works when the polynomial coefficients are all 1.
If we normalize the polynomial for each recursion we get a complicated result. But consider each coefficient in the from r^(cn/n) for n = 1,Nmax. That makes the the polynomial have a dot product of Pk(n)  dot Ck(N), where the Ck are the sequenc of coefficients written as powers of the base,r. That limits out set of coefficients.
But the coefficients set a bound on error if the error functions obeys a triangle rule.  So the error bound becomes 1/sum(Cn^(1/n)) fior n = Nmax to 1. Just use that error bound and we get a hyperbolic with the narrow angle.


This works because the finite log is always a rational fraction.  So when the coefficients narrow the error bounds, the integer p and q making p/q become very large.