The Distribution of the Cusped Hypocycloidal Mahler Measure
| 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.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.identifier.uri | https://hdl.handle.net/1794/27748 | |
| 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.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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