TITLE: Rational exponential expressions
QUESTION [7 upvotes]: Consider the following extension of polynomials.  The rational exponential expressions (REXes) are given by:

The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary functions of addition, multiplication, exponentiation, and division.

This is a restriction to positive constants of a class of expressions studied in Brown, 1969, "Rational Exponential Expressions and a Conjecture Concerning π and e".  Buchberger & Loos, 1982, "Algebraic Simplification", sect. 6, mention the existence of algorithms for finding canonical forms for these expressions, but say their correctness depends on number-theoretic conjectures that have not been settled.
I'm interested in a decision procedure for dominance, where for two REXes in one variable, $f$ and $g$, $f \leq g$ if $\exists N \in {\mathbb N}.\forall n \in {\mathbb N}. N \leq n \implies f(n) \leq g(n)$.  Do we have either the existence of canonical forms, or decidability for this problem?

REPLY [12 votes]: Yes, there are algorithms to decide asymptotic dominance - in fact for a much wider class of elementary functions. I discovered the first such algorithm circa 1980 while an undergrad member of the MIT Mathlab group researching effective algorithms for computing limits for the Macsyma symbolic computation system. Another different algorithm was discovered independently a handful of years later by John Shackell. You should be able to find references to the literature by googling the more recent buzzword "transseries".
Many computer algebra systems have (partial) implementations of these algorithms. Here's an example of my algorithm on (my generalization of) an example Rich Schroeppel proposed to attempt to stump my algorithm (he was convinced no such algorithm existed). It shows that $\rm{\: lim_{x\to\infty}\ d40 = e^a}$

Don't dare try L'Hopital's rule on that monster!
For further discussion (and the text form of the above image) see my post on sci.math, 1996/03/20, L'Hopital's rule question:
http://groups.google.com/group/sci.math/msg/05298104ac44efd2
http://groups.google.com/groups?selm=WGD.96Mar20231913%40berne.ai.mit.edu