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(Quoted from the article) Our object is the theory of "{\pi}-exponentials"Pulita developed in his thesis [...] We start with an abstract algebrastatement about the structure of the kernel of iterations of the Frobeniusendomorphism on the ring of Witt vectors with coordinates in the ring ofintegers of an ultrametric extension of $\mathbf{Q}_p$. Provided sufficiently(ramified) roots of unity are available, it is, unexpectedly simply, aprincipal ideal with respect to an explicit generator essentially given byPulita's {\pi}-exponential. This result is a consequence and a reformulation ofcore facts of Pulita's theory. It happened to be simpler to prove directly thanreformulating Pulita's results. Its translation in terms of series is veryelementary, and gives a criterion for solv- abilty and integrality for p-adicexponential series of polynomials. We explain how to deduce an explicit formulaof their radius of convergence, and even the function radius of convergence. Werecover this way, in elementary terms, with a new proof, and importantsimplifications, an algorithm of Christol based similarly on Pulita's work. Oneconcrete advantage is: one can easily prove rigorous complexity bounds aboutthe implied algorithm from our explicit formula. We also add there and thererefinements and observation, notably hinting some of the finer informationsthat can also given by the algorithm. One of the appendix produce a computationwhich gives finer estimates on the coefficients of these series. It shouldprovide useful in proving complexity bounds for various computational useinvolving these series.[...]

Sur les {\pi}-exponentielles I : Vecteurs de Witt annul\'es par Frobenius et Algorithme de (leur rayon de convergence)