dc.contributor.advisor |
Sinclair, Christopher |
|
dc.contributor.author |
Hunter, Nathan |
|
dc.date.accessioned |
2022-10-26T15:26:47Z |
|
dc.date.available |
2022-10-26T15:26:47Z |
|
dc.date.issued |
2022-10-26 |
|
dc.identifier.uri |
https://scholarsbank.uoregon.edu/xmlui/handle/1794/27748 |
|
dc.description.abstract |
We explore generalized Mahler measures associated to regions in the complex plane. These generalized Mahler measures describe the complexity of polynomials in C[x] by comparing the geometry of their roots to subsets of C. Citing past work connecting the Mahler measure to the unit disk and the reciprocal Mahler measure to the interval [-2,2], we explore a family of cusped hypocycloidal Mahler measures mu^(N) associated to the (N+1)-cusped hyplocycloids, using potential theory to show how a generalized Mahler measure may be constructed from Jensen's formula.Let s be a complex variable, and d a positive integer. To every generalized Mahler measure Phi we define the complex moment function Hd(Phi; s) which provides information about the range of values Phi takes on degree d polynomials in C[x]. These functions are analytic in the half-plane R(s)>d. We will show how Hd(s) may be represented as the determinant of a Gram matrix in a Hilbert space determined by Phi and s. We thus discover properties of Hd(mu^(N); s) as a rational function of s. |
en_US |
dc.language.iso |
en_US |
|
dc.publisher |
University of Oregon |
|
dc.rights |
All Rights Reserved. |
|
dc.title |
The Distribution of the Cusped Hypocycloidal Mahler Measure |
|
dc.type |
Electronic Thesis or Dissertation |
|
thesis.degree.name |
Ph.D. |
|
thesis.degree.level |
doctoral |
|
thesis.degree.discipline |
Department of Mathematics |
|
thesis.degree.grantor |
University of Oregon |
|