Waring's Problem for Linear Polynomials and Laurent Polynomials
Kim, Dong-Il
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https://hdl.handle.net/2142/86812
Description
Title
Waring's Problem for Linear Polynomials and Laurent Polynomials
Author(s)
Kim, Dong-Il
Issue Date
2003
Doctoral Committee Chair(s)
Aimo Hinkkanen
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Waring's problem is about representing any function in a class of functions as a sum of kth powers of nonconstant functions in the same class. We allow complex coefficients in these kind of problems. Consider i=1p1 fiz k=z and i=1p1 fizk =1 . For a given k ≥ 2, let p 1 and p2 be the smallest numbers of functions that give the above identities. W. K. Hayman obtained lower bounds of p1 and p2 for polynomials, entire functions, rational functions and meromorphic functions. First, we consider Waring's problem for linear polynomials and get p 1 = k and p2 ≥ k + 1. Next, we study Waring's problem for Laurent polynomials and obtain lower bounds of p1 and p 2. Finally, we discuss the misquote that I discovered in the proof of Hayman's theorem.
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