Nevanlinna Theory and Rational Values of Meromorphic Functions

Chalebgwa, Taboka Prince (2019-04)

Thesis

ENGLISH ABSTRACT: In this thesis, we are concerned with the problem of counting algebraic points of bounded height and degree on graphs of certain transcendental holomorphic and meromorphic functions. Adopting a Nevanlinna theoretic approach for the latter, we attain bounds of the form C(d)(log H)b for the number of algebraic points of height at most H and degree at most d on the restrictions to compact subsets of domains of holomorphy of meromorphic functions with certain growth/decay conditions. In the second half of the thesis, we turn our attention to counting points on graphs of certain analytic functions with growth behaviour stricter than finite order and positive lower order. For these functions, we are able to relax the need to restrict them to compact subsets of C, and indeed, to count points either on the whole graph or nearly all of it. For these functions we also attain a bound of the form C(d)(log H)h. We end this work with several pointers towards possible extensions of our results. The results in this thesis can be seen as extensions of the work of Boxall and Jones on algebraic values of certain analytic functions.

AFRIKAANSE OPSOMMING: In hierdie tesis is die probleem om die algebraïese punte van begrensde hoogte en graad op grafieke van sekere transendentale holomorfiese en meromorfiese funksies te tel, van belang. Met behulp van die Nevanlinnateoretiese benadering vir laasgenoemde, verkry ons grense van die vorm C(d)(logH)b vir die getal algebraïese punte waarvan die hoogte op die meeste H en die graad op die meeste d is, met die beperkings tot kompakte deelversamelings van domeine van holomorfie van meromorfiese funksies met sekere groei-/verval-voorwaardes. In die tweede helfte van die tesis vestig ons ons aandag op die tel van punte op grafieke van sekere analitiese funksies met groei-gedrag strenger as eindige orde en positiewe onderorde. Vir hierdie funksies kan ons die beperking tot kompakte deelversamelings van C ophef en, inderdaad, die punte op óf die hele grafiek, óf byna die hele grafiek, tel. Vir hierdie funksies verkry ons ook a grens van die vorm C(d)(logH)h. Ons sluit hierdie werk af met verkeie aanduidings van moontlike uitbreidings van ons resultate. Die resultate in hierdie tesis kan as uitbreidings van die werk van Boxall en Jones oor algebraïese waardes van sekere analitiese funksies, beskou word.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/108683
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