Des $\pi$-exponentielles I: vecteurs de Witt annulés par Frobenius et algorithme de (leur) rayon de convergence

Abstract

Our object is the theory of “$\pi$-exponentials” Pulita developed in his thesis, generalising Dwork's and Robba's exponentials and extending Matsuda's work:

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 ${\mathbb{Q}}_p$. 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 solvabilty 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. It is not apparent yet in the present work, but should be in latter projected developments, the series under consideration are the base object for some exponential sums on finite fields via $p$-adic approach, namely via rigid cohomology with rank one coefficients.

Convergence radius and coefficients estimates are involved studying the efficiency of computational implementations of these objects. The understanding of convergence radius of $p$-adic differential equations is a subject undergoing active developments, and we here provide a fine theoretical and computational study of the simplest of cases.

This initiate a projected series of articles. We start here, with the case of the affine line as a base space, a Witt vectors paradigm. This provides an alternative purely algebraic approach of Pulita's theory; the richness of Witt vectors theory allow suppleness and efficiency in working with $\pi$-exponentials, which will prove efficient later in the series.