On Diophantine exponents for Laurent series over a finite field

Abstract

In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j\geq 1}$ and a sequence of algebraic Laurent series $(\xi_j)_{j\geq 1}$ such that deg $\xi_j=p^{k_j}+1$ and \begin{equation} w_1(\xi_j)=w_1 ^{*}(\xi_j)=\ldots =w_n(\xi_j)=w_n ^{*}(\xi_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{*}(\xi )$ are different.Comment: 22 page

Mahler’s and Koksma’s Classifications in Fields of Power Series

“…For , Ooto [19] proved the existence of in for which . His strategy, inspired by [6], was to use continued fractions to construct power series with and sufficiently large to ensure that, for small (in terms of ) values of n , we have Very recently, Ayadi and Ooto [2] answered a question of Ooto [20, Problem 2.2] by proving, for given and , the existence of algebraic power series in for which .…”

Mahler's and Koksma's classifications in fields of power series

“…Do there exist power series ξ in F q ((T −1 )) such that w * n (ξ) < w n (ξ), for infinitely many n? The formulation of the next problem is close to that of [21,Problem 2.4]. Problem 6.3.…”

On quadratic approximation for hyperquadratic continued fractions

“…[4,11,16,17]) and algebraic approximations (e.g. [14,18,19]) for algebraic Laurent series. In particular, the second author ( [14]) proved that, for any rational number w > 2n − 1, there exists an algebraic Laurent series α ∈ F q ((T −1 )) such that w 1 (α) = w * 1 (α) = .…”

“…. = w n (α) = w * n (α) = w. The first purpose of this paper is to consider Problem 2.2 in [14]. Problem 1.1.…”