Analytic reparametrization of semi-algebraic sets and local complexity bounds
In many problems in Analysis and Dynamics it is important to subdivide objects under consideration into simple pieces, keeping control of high order derivatives. It is known that semi-algebraic sets $A$ inside the unit ball allow for a $C^k$ - triangulation, where each simplex is represented as an image, under the ``reparametrization mapping" $\psi$, of the standard simplex, with all the derivatives of $\psi$ up to order $k$ bounded by $1$. The number of simplices in this triangulation is bounded through the degree of $A$.
The main result presented in this talk is, that if we reparametrize all the set $A$ but its small part of a size $\delta
> 0$, we can do much more: not only to ``kill" all the
derivatives at once, but to bound uniformly the {\it analytic complexity} of the pieces, while their number remains of order $log \ ({1\over \delta})$. In contrast with the $C^k$-case, the number of pieces in an {\it analytic} reparametrization {\it cannot be bounded through the degree only}, and the above result is, essentially, sharp.
Relations to the Bernstein type inequalities for algebraic functions, as well as some new open questions concerning the complexity of semi-algebraic sets will be stressed. Initial applications in Analytic Dynamics will be discussed, in particular, explicit bounds on the local volume growth in iterations of analytic mappings.