University of Illinois Urbana-Champaign

Schanuel Functions and Algebraic Differential Equations

Reinhart, Georg Martin

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Description

Title
Schanuel Functions and Algebraic Differential Equations
Author(s)
Reinhart, Georg Martin
Issue Date
1993
Doctoral Committee Chair(s)
Rubel, Lee A.
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
Abstract
  • The Schanuel class consists of functions that can be obtained from $\doubc$, the variable z, and closed under addition, multiplication and exponentiation by the exponential function. A proof is presented that the minimum order of an algebraic differential equation (ADE) satisfied by a given Schanuel function depends on the number of algebraically independent functions (over $\doubc (z))$ needed to build up the function. This extends a result of A. Babakhanian on towers of exponentials.
  • We also give an algorithm for determining the ADE of minimum order described above. This algorithm has been implemented on computers using Mathematica.
  • Then formal power series satisfying ADEs are investigated. The main theorem is that the length of the gaps such a power series may have, can grow at most linearly with the position of the gap. The linearity constant is determined by the degree of the ADE the power series satisfies. The algorithm mentioned above also gives an estimate on the degree of the ADE of minimum order. Thus we obtain an estimate on the length of gaps the Taylor series of a Schanuel function can have.
  • In the last chapter it is shown that if f(z) is a nonconstant entire solution of the functional equation $f(z + 1) = e\sp{f(z)},$ then f(z) cannot satisfy an ADE. This fact is also a consequence of a result of M. Boshernitzan and Lee A. Rubel, but the method is fundamentally different. Indeed, our method is combinatorial in nature and of interest in the field of partitions in its own right.
  • Finally, some new applications are given. It will be shown that a nonconstant entire function of the form $f = e\sp{f\sb1}, f\sb1 = e\sp{f\sb2},\...$ cannot satisfy an ADE. Furthermore, we give a partial answer to a question posed by Lee A. Rubel: The iterates of a nonconstant entire function w of the form w = exp (exp (exp(z))) cannot satisfy one and the same ADE.
Graduation Semester
1993
Type of Resource
text
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http://hdl.handle.net/2142/72541

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