## Distribution modulo one in a formal power series field over a finite field. (Répartition modulo 1 dans un corps de séries formelles sur un corps fini.)(French)Zbl 0819.11026

Let $$q$$ be a power of a prime number $$p$$ and let $$\mathbb{F}_ q$$ be the finite field with $$q$$ elements. Many problems in number theory have been extended to the ring $$\mathbb{F}_ q [T]$$. One of them is the problem of the uniform distribution modulo 1. The local field $$\mathbb{F}_ q ((T^{-1}))$$ of Laurent series takes the place of the field of real numbers, the valuation ideal takes the place of the interval $$[0,1[$$. It is well-known that the sequence $$(\sqrt {n})$$ is uniformly distributed modulo 1. It is natural to study the problem of the uniform distribution of the sequence $$(H^{1/2})$$, $$H$$ running through the set of polynomials in $$\mathbb{F}_ q [T]$$ admitting a square-root $$H^{1/2}$$ in the field $$\mathbb{F}_ q ((T^{-1}))$$.
More generally, let $$\ell$$ be an integer $$\geq 2$$, coprime with $$p$$. In this paper we describe polynomials in $$\mathbb{F}_ q [T]$$ admitting an $$\ell$$-th root in the field $$\mathbb{F}_ q ((T^{-1}))$$ and define a map $$H\mapsto H^{1/\ell}$$ from the set $${\mathcal L}$$ of these polynomials to the set $$\mathbb{F}_ q ((T^{-1}))$$. We prove that the sequence $$(H^{1/\ell})$$, $$H$$ running through the set $${\mathcal L}$$, is uniformly distributed modulo 1, and that for $$\ell \geq 3$$, the sequence $$(P^{1/\ell})$$ is uniformly distributed modulo 1, $$P$$ running through the set of irreducible polynomials of $${\mathcal L}$$.
Reviewer: M.Car (Marseille)

### MSC:

 11K41 Continuous, $$p$$-adic and abstract analogues 11T55 Arithmetic theory of polynomial rings over finite fields 11K06 General theory of distribution modulo $$1$$ 11T06 Polynomials over finite fields 12J10 Valued fields
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