Leia Molin: Nollställen till slumppolynom: när och varför hopar de sig kring enhetscirkeln?

Abstract:

The purpose of this thesis is to explore an interesting phenomenon concerning the distribution of zeroes of random polynomials with independent coefficients. The starting point is a simulation where we create a random polynomials coefficients were independent and uniformly distributed. If we then determine the roots numerically and do a visual representation we see that the roots seem to closter near the unit circle. A theoretical overview of some basics of complex analysis is given, and we attempt to interpret this figure and answer the question why it appears as it does.

The first part of the thesis contains relevant probabilistic and analytic definitions. The second segment is centered on Jensen’s formula. The formula is proved by using Cauchy’s integral formula, the mean value property and the proof to why the complex logarithm is holomorphic. Jensen’s formula is important because it supplies a method for estimating the number of zeroes of an analytic function; and is thus directly linked to our circle and purpose of this thesis.

In the last segment of this thesis we summarize some key aspects of Nikeghbali and Hughes’ article The Zeroes of random Polynomials cluster uniformly near the unit circle, and furthermore use the material from the thesis to solve a smaller problem.