Distribution of Critical Points of Polynomials

• Main Author: Forkéus, Ted
• Format: Others
• Language: English
• Published: Karlstads universitet 2021
• Subjects: Polynomials; Critical points; Probability theory; Analysis; Sendov's conjecture; Probability Theory and Statistics; Sannolikhetsteori och statistik; Mathematical Analysis; Matematisk analys
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-82941

This thesis studies the relationship between the zeroes of complexpolynomials in one variable and the critical points of those polynomials. Our methods are both analytical and statistical in nature, usingtechniques from both complex analysis and probability theory. Wepresent an alternative proof for the famous Gauss-Lucas theorem aswell as proving that the distribution for the critical points of a randompolynomial with real zeroes will converge in probability to the distribution of the zeroes. A simulation of the case with complex zeroesis also presented, which gives statistical support that this holds forrandom polynomials with complex zeroes as well. Lastly, the previous results are then applied to Sendov’s conjecture where we take aprobabilistic approach to this problem.