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 $(ξ_j)_{j\geq 1}$ such that deg $ξ_j=p^{k_j}+1$ and \begin{equation} w_1(ξ_j)=w_1 ^{*}(ξ_j)=\ldots =w_n(ξ_j)=w_n ^{*}(ξ_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $ξ$ for which $w_n(ξ)$ and $w_n^{*}(ξ)$ are different.