Random Polynomials and Approximate Zeros of Newton's Method
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In this paper we study the area of approximate zeros for Newton's method, i.e. the set of points for which Newton's method converges doubly exponentially fast starting from the first iterate. We obtain a bound in terms of the separation of the roots. We then apply this various probability distributions on polynomials of fixed degree, obtaining estimates on the probability that the approximate zero region has small area. For polynomials of the form $f(z)=\sum_{i=0}^d a_iz^i$ with $a_d=1$ and complex valued $a_i$ chosen independently and uniformly in the unit ball for $id^{-2-\eps}$ for any $\eps>0$.