  (Quoted from the article) Our object is the theory of "{\pi}-exponentials"
Pulita developed in his thesis [...] We start with an abstract algebra
statement about the structure of the kernel of iterations of the Frobenius
endomorphism on the ring of Witt vectors with coordinates in the ring of
integers of an ultrametric extension of $\mathbf{Q}_p$. Provided sufficiently
(ramified) roots of unity are available, it is, unexpectedly simply, a
principal ideal with respect to an explicit generator essentially given by
Pulita's {\pi}-exponential. This result is a consequence and a reformulation of
core facts of Pulita's theory. It happened to be simpler to prove directly than
reformulating Pulita's results. Its translation in terms of series is very
elementary, and gives a criterion for solv- abilty and integrality for p-adic
exponential series of polynomials. We explain how to deduce an explicit formula
of their radius of convergence, and even the function radius of convergence. We
recover this way, in elementary terms, with a new proof, and important
simplifications, an algorithm of Christol based similarly on Pulita's work. One
concrete advantage is: one can easily prove rigorous complexity bounds about
the implied algorithm from our explicit formula. We also add there and there
refinements and observation, notably hinting some of the finer informations
that can also given by the algorithm. One of the appendix produce a computation
which gives finer estimates on the coefficients of these series. It should
provide useful in proving complexity bounds for various computational use
involving these series.[...]
