User:Chenxlee/Miller

Miller's dichotomy is a result in model theory demonstrated by Chris Miller. It states that a dichotomy exists in the class of o-minimal structures expanding the ordered field of real numbers: in a given structure either every definable function grows slower than some polynomial, or else the exponential function is definable in the structure.

Statement
An o-minimal structure R = (R,<,&hellip;) expanding the real ordered field is called polynomially bounded if for every definable function &fnof;:R&rarr;R there is a real number x0 and a natural number n such that for any x &gt; x0 it is the case that
 * $$|f(x)|\leq x^n.$$

Not every structure R is polynomially bounded: the exponential function exp(x) eventually grows faster than any polynomial, so the structure formed by adding the exponential function to the real ordered field is not polynomially bounded. That this structure is o-minimal is a consequence of Wilkie's theorem.

Miller proved the surprising result that if a structure R is not polynomially bounded then the exponential function is definable in R. Since this function can't be defined in any polynomially bounded structure this statement is in fact an if and only if one, hence partitioning the class of o-minimal structures into those that are polynomially bounded and those that define the exponential function.

Proof of the statement
The proof uses results on Hardy fields to show that in non-polynomially bounded structures one can construct a definable function g whose derivative is asymptotic to x&minus;1. From the existence of this function one can infer the definability of the natural logarithm, using which it is simple to define the exponential function.

Extensions
The dichotomy as stated above applies only to o-minimal expansion of the real ordered field, but the result was later extended mutatis mutandis to arbitrary real closed fields. Specifically Miller proved that o-minimal structures in this more general class satisfy a property called power boundedness, which generalises polynomial boundedness, or else they define a non zero function &fnof; that is equal to its derivative.