Geometry of Complex Polynomials: On Sendov's Conjecture
| dc.contributor.advisor | Boxall, Gareth John | en_ZA |
| dc.contributor.advisor | Breuer, Florian | en_ZA |
| dc.contributor.author | Chalebgwa, Taboka Prince | en_ZA |
| dc.contributor.other | Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences | en_ZA |
| dc.date.accessioned | 2016-12-22T13:13:52Z | |
| dc.date.available | 2016-12-22T13:13:52Z | |
| dc.date.issued | 2016-12 | |
| dc.description | Thesis (MSc)--Stellenbosch University, 2016 | en_ZA |
| dc.description.abstract | ENGLISH ABSTRACT : Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each α ε (0, 1), there is an integer N such that for any polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside the unit disk, the disk │z- α│ ≤ 1 contains a critical point of P(z). Basing on this result, we derive an explicit formula N(a) for each α ε (0, 1) and, furthermore, obtain a uniform bound N for all a ε [α,β] where 0 < α < β < 1. This addresses the questions posed in Dégot’s paper. | en_ZA |
| dc.description.abstract | AFRIKAANSE OPSOMMING : Die vermoede van Sendov lui dat, as alle nulpunte van ’n komplekse polinoom P(z) van graad minstens twee binne die eenheidssirkel lê, dan is daar ’n kritieke punt van P(z) binne ’n afstand van een van elke nulpunt. In die artikel wat 2014 verskyn het, het Dégot bewys dat daar vir elke a ε (0, 1) ’n heelgetal N bestaan sodat, vir elke polinoom P(z) van graad groter as N met P(a) = 0 en met alle nulpunte binne die eenheidskyf, die skyf │z- α│≤1 ’n kritieke punt van P(z) bevat. Gebaseer op hierdie werk bepaal ons ’n formule N(a) vir elke a ε (0, 1), en verder bepaal ons ’n uniforme bogrens N vir alle a ε [α,β] waar 0 < α < β < 1. Dit spreek die vrae aan wat in Dégot se artikel gestel is. | af_ZA |
| dc.format.extent | vi, 74 pages : illustrations | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/10019.1/100088 | |
| dc.language.iso | en_ZA | en_ZA |
| dc.publisher | Stellenbosch : Stellenbosch University | en_ZA |
| dc.rights.holder | Stellenbosch University | en_ZA |
| dc.subject | Sendov's conjecture | en_ZA |
| dc.subject | Complex polynomials | en_ZA |
| dc.subject | Geometry -- Conjectures | en_ZA |
| dc.subject | Gauss-Lucas theorem | en_ZA |
| dc.title | Geometry of Complex Polynomials: On Sendov's Conjecture | en_ZA |
| dc.type | Thesis | en_ZA |