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

Car, Mireille

Acta Arith. 69, No. 3, 229-242 (1995).

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

Keywords:

uniform distribution; polynomials; finite fields; Weyl criterion; field of formal Laurent series; valuation