The Distribution of the Cusped Hypocycloidal Mahler Measure
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Date
2022-10-26
Authors
Hunter, Nathan
Journal Title
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Publisher
University of Oregon
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.