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Geometry of Complex Polynomials: On Sendov's Conjecture
Thesis (MSc)--Stellenbosch University, 2016
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2016
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| _version_ | 1867613882863845376 |
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| access_status_str | Open Access |
| author | Chalebgwa, Taboka Prince |
| author2 | Boxall, Gareth John |
| author_browse | Boxall, Gareth John Chalebgwa, Taboka Prince |
| author_facet | Boxall, Gareth John Chalebgwa, Taboka Prince |
| author_sort | Chalebgwa, Taboka Prince |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2016 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/100088 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:43:12.690Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/100088 Geometry of Complex Polynomials: On Sendov's Conjecture Chalebgwa, Taboka Prince Boxall, Gareth John Breuer, Florian Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences Sendov's conjecture Complex polynomials Geometry -- Conjectures Gauss-Lucas theorem Thesis (MSc)--Stellenbosch University, 2016 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. 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. 2016-12-22T13:13:52Z 2016-12-22T13:13:52Z 2016-12 Thesis http://hdl.handle.net/10019.1/100088 en_ZA Stellenbosch University vi, 74 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Sendov's conjecture Complex polynomials Geometry -- Conjectures Gauss-Lucas theorem Chalebgwa, Taboka Prince Geometry of Complex Polynomials: On Sendov's Conjecture |
| title | Geometry of Complex Polynomials: On Sendov's Conjecture |
| title_full | Geometry of Complex Polynomials: On Sendov's Conjecture |
| title_fullStr | Geometry of Complex Polynomials: On Sendov's Conjecture |
| title_full_unstemmed | Geometry of Complex Polynomials: On Sendov's Conjecture |
| title_short | Geometry of Complex Polynomials: On Sendov's Conjecture |
| title_sort | geometry of complex polynomials on sendov s conjecture |
| topic | Sendov's conjecture Complex polynomials Geometry -- Conjectures Gauss-Lucas theorem |
| url | http://hdl.handle.net/10019.1/100088 |
| work_keys_str_mv | AT chalebgwatabokaprince geometryofcomplexpolynomialsonsendovsconjecture |