Graduation Year
2026
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Seung-Yeop Lee, Ph.D.
Committee Member
Dmitry Khavinson, Ph.D.
Committee Member
Evgenuii Rakhmanov, Ph.D.
Committee Member
Razvan Teodorescu, Ph.D.
Committee Member
Sergey Lisenkovf, Ph.D.
Keywords
Coulomb Gas, Planar Orthogonal Polynomials, Riemann-Hilbert Problems
Abstract
Planar orthogonal polynomials of degree n are monic polynomials defined by the orthogonality relation,
∫cPn(z)Pm(z)e-V(z) dA(z) = hnδn,m.
where the integration is over the whole complex plane with Lebesgue area measure dA, and hn is a positive norming constant. In this thesis we consider the potential V: CR of the type,
V(z) = |z|2p-2Re Σm j=1 tjzj
we characterize the planar orthogonal polynomials using novel matrix Riemann-Hilbert problems. We pro-pose a Riemann-Hilbert problem of size 2px 2p when the maximal degree satisfies m ≤ p. For m = 2p, we propose a Riemann-Hilbert problem of size 3px 3p for the related bi-orthogonal polynomials.
Scholar Commons Citation
Arenas, Abril, "Riemann-Hilbert Problems for a Class of Planar Orthogonal Polynomials" (2026). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/11232
