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# Sums of rational functions

Author:

Date created:

2012-04-12

Identifier:

etd7229

Keywords:

Number Theory

Function fields

Algebra

Automatic sequences

Abstract:

Given a rational function φ(X) with rational coefficients that is defined at every positive integer, we consider the sum of φ(n) as n runs from 0 to infinity. It is believed that when this sum converges, it converges to either a rational or transcendental number. We prove an analogue of this conjecture over fields of rational functions: Let K be a field and let φ(X) be a rational function with coefficients in K such that φ(0) = 0. Given a positive integer d ≥ 2, we define F(X) to be the sum of φ(X^(d^n)) as n runs from 0 to infinity. If d is not a power of char(K), then F(X) is either a rational function or transcendental over K(X). Our demonstration uses results from the theory of automatic sequences and from commutative algebra.

Document type:

Thesis

Rights:

Copyright remains with the author. The author granted permission for the file to be printed and for the text to be copied and pasted.

File(s):

Senior supervisor:

Jason Bell

Department:

Science: Department of Mathematics

Thesis type:

(Thesis) M.Sc.

Statistics: