Simplification of exponential factors of irregular connections on $\mathbb P^1$

We give an explicit algorithm to reduce the ramification order of any exponential factor of an irregular connection on $\mathbb P^1$, using the same types of basic operations as in the Katz-Deligne-Arinkin algorithm for rigid irregular connections. The exponential factor reached when the algorithm terminates is, up to admissible deformations, the unique factor with minimal ramification order in the orbit of the initial factor under successive applications of basic operations. Furthermore, we show that for every even integer $n\geq 0$, there is up to admissible deformations a finite number of non-simplifiable exponential factors at infinity such that the corresponding elementary wild character variety has complex dimension $n$, which conjecturally implies that there is a finite number of isomorphism classes of elementary wild character varieties in any dimension. These results can be viewed as saying that the set of all possible level data of exponential factors has the structure of a disjoint union of an infinite number of infinite rooted trees, each tree being associated to a given dimension $n$ and with a finite number of trees for each $n$. In particular, in dimension 2 there is a unique tree, corresponding to the Painlev\'e I moduli space.