Mild parametrizations
This is a summary about my results on parametrizations of power-subanalytic sets via an elementary and well known example.
Copyright © International GeoGebra Institute, 2021
Mild parametrization of
Yomdin's Example
Yomdin's Example
Consider the family of semi-algebraic curves defined by:
where t<x<1, and t is a parameter in (0,1). The goal is to cover the graph of this function, which you can see on the left, by finitely many (independent of t) functions with domain (0,1) that are "mild". This means that
for some real positive A,B and C, where n is at most some desired order r. Yomdin used this example to show that one cannot take C = 0 (in the case r is infinite). In that case, the function would be analytic.
C1 parametrization
We will cover the graph of this family of curves with 3 functions that have bounded C1-norm, which means that they are mild up to order 1 if you want. This is called a C1 parametrization. At the moment, we obtain a C0 parametrization consisting of 1 function after rescaling the domain of the function above. This yields the function
To obtain the C1 parametrization, we just check where the first order derivative of the original function t/x is not bounded (in absolute value) by some value B, let's say B = 1. Thus, we compute
This condition divides our domain into three parts, one being a point, see the figure on the right. The point P will be parametrized by a constant map. The other two parts are symmetric. This observation leads to the following three functions.
Copyright © International GeoGebra Institute, 2021
Cr parametrization
From the parametrization above, we will deduce a parametrization with maps that are mild up to order r. It is easily deduced from the C1 parametrization by simply substituting a power of x, in this case r, for x.
One can then bound the Cr-norm by performing a suitable linear coordinate change and obtain a number of charts that is (roughly) just r.
Mild parametrization
To deduce a mild parametrization (up to order infinity), we will substitute
into the three functions above. In that case the second and third function become mild, where C = 2.
Since we are working with a family of curves, it is possible to change the order of operations we have performed here to obtain C = 1. The details can be found in this article.
Mild parametrizations of power-subanalytic sets
One can generalize the results above to bounded power-subanalytic sets, uniform in families. The first step is to find functions describing this family, which are obtained by cell decomposition (the structure of power-subanalytic sets is o-minimal). Next, one uses a preparation theorem and a change of coordinates (if necessary) to obtain maps that are monomial (times a unit) and have bounded C1-norm. Finally, one can then apply the substitutions as above. This "algorithm" produces a number of charts that is polynomial in r of degree dim(X), where X is the set that is being parametrized.